SB
MATHEMATICAL STUDIES: 1
TRANSFORMATION
GEOMETRY
MAX JEGER
M
GEORGE
ALLEN
AND
UN WIN
In 1872 Felix Klein, speaking at the University of ErJangen,
suggested that various geometries should be distinguished by
the groups of transformations under which their propositions
remain valid. This exciting and important idea has had many
repercussions in the world of mathematics and recently its
effects have been felt in the school classroom: an outstanding
feature of new mathematics syllabuses being the inclusion of an
approach to geometry based on the study of such plane trans-
formations as rotations and reflections. This study is doubly
profitable, for not only do transformations help to throw
geometric properties into sharp relief, but they also provide
a fascinating introduction to group theory. Both of these
aspects are given due consideration in Mr Jeger's book, which
was described in Mathematics Teaching as 'perhaps the best
development of school geometry from the group point of view
which is to be found anywhere'.
Readers new to this type of geometry will be surprised by the
power and versatility of transformations and by the way that
they can be used to solve many different types of construction
problems in addition to such well-known results as the nine-
point circle. The second aspect is of equal interest. The ways in
which transformations are related and the groups they form
are investigated. It is shown how the reflections generate ail the
other congruence-preserving transformations, and that if, in
addition, enlargements are considered, the result is a new group
—the group of similarities which characterizes Euclidean
geometry. Finally, a look is taken at the geometry associated
with affine transformations.
This book is very suitable for sixth-formers and under-
graduates who want a not too abstract approach to groups,
for teachers who want a compact and workable account of this
kind of geometry, and for training college students as an intro-
duction to new ideas in algebra and geometry.
Jacket design by Timothy Drever
Transformation Geometry
Max Jeger
George Allen & Unwin Ltd,
Price in U.K. only: 25s. net
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MATHEMATICAL STUDIES
A Series for Teachers and Students
EDITED BY
DAVID WHEELER
University of Leicester
No. 1
TRANSFORMATION GEOMETRY
In the same series
No. 2
ALGEBRAIC STRUCTURES
by A. W. Bell
In preparation
topology by J. E, Reeve
vectors by E. H. Leaton
logic by W. O, Storer
Transformation
Geometry
By
MAX JEGER
Oberrealschule, Lucerne
ENGLISH VERSION BY
A. W. DEICKE AND A.
University of Southampton
G. HOWSON
yi.-H
London
GEORGE ALLEN AND UNWIN LTD
RUSKIN HOUSE MUSEUM STREET
FIRST PUBLISHED IN ENGLISH IN 1966
FOREWORD
This book is copyright under the Berne Convention.
Apart from any fair dealing for the purposes of
private study, research, criticism or review, as
permitted under the Copyright Act, 1956, no
portion may be reproduced by any process without
written permission. Enquiry should be made to
the publisher.
This translation © George Allen & Unwin Ltd., 1966
Translated from the German
KONSTRUKTIVE ABBILDUNGSGEOMETRIE
© 1964 Raber Vcrlag Lucerne and Stuttgart
PRINTED IN GREAT BRITAIN
in 10 point Times Roman type
BY C. TINLING AND CO. LTD.,
LIVERPOOL, LONDON AND FRESCOT
It is generally agreed that school mathematics syllabuses are
in need of reform. The traditional syllabus is no longer an ade-
quate preparation for mathematics as it is taught at a higher
level ; it indicates very little of the range of contemporary uses
of mathematics; and it contains a high proportion of routine
computation and manipulation at the expense of mathematical
ideas which yield immediate enjoyment and satisfaction. A
number of schools are now experimenting with new syllabuses
which attempt to cure these faults.
Whether the experiments prove to be wholly successful or not,
they are bringing a new element into the situation: an awareness
that it is part of the job of the teacher of mathematics to inform
himself about the relatively recent developments and changes in
his subject. It is no longer possible to believe that developments
in mathematics concern only the research mathematician and
do not have any bearing on the mathematics taught in schools.
This series of books is intended as a contribution to the reform
of school mathematics by introducing to the reader some areas
of mathematics which, broadly speaking, can be called modern,
and which are beginning to have an influence on the content of
school syllabuses.
The series does not put forward explicit advice about what
mathematics to teach and how it should be taught. It is meant to
be useful to those teachers and students in training who want to
know more mathematics so that they can begin to take part in
the existing experimental schemes, or modify them, or devise
their own syllabus revisions, however modest, The books are
elementary without being trivial : the mathematical knowledge
they assume is roughly that of a traditional grammar school
course, although substantial sections of all the books can be
understood with less.
Now that the stability over a long period of school mathe-
matics syllabuses seems to be corning to an end, it is to be hoped
that a new orthodoxy does not succeed the old. The reform of
mathematics teaching should be a continuing process, associ-
ated with a deepening study of the subject throughout every
teacher's professional life. These books may help to start some
teachers on that course of study. d.w.
PREFACE TO THE THIRD GERMAN EDITION
The teaching of mathematics in grammar schools today is in
an unsettled state. Transformation geometry, which has recently
been introduced into school syllabuses, is the outcome of a
series of reforms which have aimed at liberating the teaching of
geometry from domination by Euclid.
The traditional teaching of geometry is based on Euclid in
two ways. First, logic always takes precedence with Euclid ; the
arrangement of the material is essentially determined only by
logic. Theorems are put next to each other if an abstract logical
proof is possible, without regard to whether they belong
together, or whether they are essential or unimportant.
Secondly, the Euclidean style in teaching is distinguished by
its emphasis on rigid figures; rigid congruence of triangles is
considered to be the main method of proof in elementary
geometry.
The notion of transformation gives to modern teaching of
geometry a central concept comparable to the idea of function
in analysis. The hierarchy of groups of transformations leads
to a natural order for the material, and instead of the static
treatment of Euclid we have a dynamic geometry.
Although transformation geometry dates back to Felix
Klein (1849-1925), this reform is still fully in keeping with
modern tendencies which demand that, in our teaching, more
emphasis be placed on mathematical structures and isomor-
phisms. Only in one respect can it be called old-fashioned — it
seeks to preserve geometry as an essential part of mathematical
teaching.
The first two editions of this book produced a series of hints
and proposals for furthering the use of transformation geometry
in the classroom. Their aim was to make a larger circle of
teachers familiar with these new ideas. The third edition has
been completely revised. It has been extended into a course on
transformation geometry which has been tried out several times
during the last few years by the author in his own teaching.
The general arrangement of the former editions, which built up
theory and applications from concrete examples, has been
preserved. The range of problems has been considerably
extended.
PREFACE TO THE THIRD GERMAN EDITION
Reforms will only prove acceptable if they leave open various
ways of access from previous teaching. For this reason a too
rigidly systematic representation has been avoided. This course
presupposes some previous knowledge of geometry; it is
assumed that the pupil knows the elementary parts of congru-
ence and similarity geometry.
Our book differs to some degree from previous school books
on transformation geometry since it makes much more use of
the group-theoretic structure of elementary geometry. Alge-
braic operations for transformations are introduced at an early
stage and are afterwards used extensively as a method of proof.
We should like to throw open to discussion the question 'to
what extent is this algebraic approach to school geometry
likely to give the pupil more insight into the subject?' In the
final sections an attempt has been made to build a bridge from
constructive geometry to analytic geometry.
The axiomatic system behind this course has, intentionally,
not been discussed consistently. The author is convinced that in
school teaching axiomatic discussions can at most have a local
or retrospective character. But in the latter respect they belong
to the advanced sixth form stage.
The diagrams have been drawn by two pupils, A. Schenk and
R. Ronchetti, from Lucerne. Dr. H. Loeffel has helped by
reading the proofs. The author is most grateful to all these
helpers.
Lucerne, September 1963.
CONTENTS
page
FOREWORD
vii
PREFACE TO THIRD GERMAN EDITION
is
TRANSLATORS' note
xiii
LIST OF SYMBOLS
XV
1.
Mappings
17
2.
Reflection in a Line
20
3.
Translations
Combination of translations
Groups
Further example of groups
33
4. Rotations 47
Combination of rotations
5. The Group of Isometries 66
The role of reflections and half-turns in the group
of isometries
6. The Group of Transformations Mapping a
Square onto itself 75
7. Enlargements 82
The group properties of enlargements
Subgroups of the group of enlargements
8. Similarities 98
Direct similarities
Opposite similarities
9. Affine Transformations 1 10
Properties of perspective affinities
The group properties of perspective affinities
Affine geometry
10. The Affine Geometry of the Ellipse 132
BIBLIOGRAPHY 141
TRANSLATORS* NOTE
A bibliography of texts written in English on transformation
geometry would hardly impress the casual reader by its length.
Yet, although books on the subject are not numerous, there is
already a wealth of notation and nomenclature available to
budding authors. White preparing this version we have con-
stantly been faced with the need to choose between alternative
names for a particular transformation and the choice has not
always been easy. For example, 'half-turn* is an admirable
description of a rotation through 180°, but by adopting
this name one loses the line-point analogy and, what is
perhaps more important, a means of considering the trans-
formation in a way that can be easily generalized to three
dimensions. In this instance, as elsewhere, we have used the
term which is most likely to have been encountered by school-
teachers. The general similarities are not so well known as the
isometries and here we have followed Coxeter in the use of
'spiral similarity*. Since we have preferred 'enlargement' to
'dilatation' (or 'dilation', for spelling provides an additional
degree of freedom), we have not used 'dilative reflection' but
have added 'stretch-reflection' to the geometer's vocabulary.
We would plead in defence that the name is self-explanatory
and can be usefully compared with 'glide-reflection'.
One major change in notational practice has been made for
this version. Textbooks recently published in England have
tended to prefer the 'functional order* for the product of trans-
formations, that is, TR is to be interpreted as first the rotation R
and then the translation T. We have followed this convention
which seems to have much to commend it.
A.D., A.G.H.
LIST OF SYMBOLS
E(S,fi) an enlargement with centre S and scale factor [i
H a half-turn about the point O
M a a reflection in the line s
H(0,9) a rotation about the point through the directed
angle
T a translation
lV(S,0,fi) a spiral similarity with centre S, angle of rotation
and scale factor (.i
Z a stretch-reflection
<&(s,z,ft) a perspective affinity with axis of affinity s, direction a
and scale factor fi
91 (A) the group of enlargements
S (B) the group of translations and rotations
93' the group of translations and half-turns
S (K) the group of plane isometries
Q (Q) the group of affine transformations
Q a the group of perspective affinities with the given
axis s
S (S) the group of spiral similarities
S* the group of similarities
% (T) the group of translations
MAPPINGS
We introduce the idea of a mapping by means of a simple
example. Select one point S in the plane. For every point A we
can find a point A' such that the line segment A A' has its
mid-point at S. This condition ensures that a unique point A' is
associated with A. Moreover, if A runs through all the points
of the plane, then so does A'. Here we have a mapping of the
plane onto itself. If we wish to give this relation a definite
direction, then we call A the original, or object, point and A' the
image point.
If for every given object point A there is exactly one image
point A', and if for every given point B' there is exactly one
original B— that is, if the mapping is unique in both directions —
then the mapping is called one-one. Our example has this
property; here the inverse mapping, that is, the mapping which
maps the image back onto the original point, is given by the
same rule of construction.
This example illustrates a very special type of mapping. The
notion of mapping, however, includes every correspondence
n 17
18
6p
TRANSFORMATION GEOMETRY
4s
2
a*
?r
Fig 2
A
A
/ \
J. h ,
between sets of geometric objects. We shall give a rough outline
of the range of possible mappings by means of a few further
examples.
Let a and a' be two planes in space. A pencil of parallel
lines (the lines being parallel neither to « nor to a'} produces a
correspondence between the points of intersection with « and
a'. This unique mapping is called a parallel projection from a
onto a'.
A pencil of parallel lines can also define a correspondence
between the points of space and the points in a plane x. We
associate with every point P in space the point of intersection of
ji with the line passing through P. This is called a parallel
projection of space onto the plane %. In this mapping every
point P in space has a unique image P'\ however, there are
infinitely many points which are mapped onto any particular
image point Q\ The mapping is uniquely defined but is not
one-one. If the parallel lines are perpendicular to the plane n,
then the mapping is called the orthogonal projection of space
onto the plane it (see Figure 2).
The sets related by a mapping may contain different types of
objects. For example, mappings between a set of points and a
set of lines, or a set of points and a set of circles are considered
in higher geometry.* When we draw plans and elevations of an
object then we are mapping points in space onto a suitable set
of triples of points in the plane.
Similar correspondences between sets are also of great
importance in other fields of mathematics. For example, the
* Examples are poles and polars and cyclographic mappings.
MAPPINGS
19
function/: x-*y defines a correspondence between the variables
x and v. Let us consider y = x 2 . This defines a correspondence
between the real numbers in the two intervals — 5<x<5 and
0^y<25. It is easily verified that this correspondence is not
one-one. However, the same function defines a one-one corres-
pondence between the sets {x:l<x<5} and {j>:l<>><25}. If
these sets of numbers are illustrated geometrically by point
sets on two lines, we can consider the function to be a mapping.
So we see that functions and mappings are related ideas.
When 24 pupils sit in a classroom with 24 chairs, then we have
a one-one correspondence between the set of pupils and the set
of chairs. Here the correspondence is between finite sets.
Mappings between finite sets also occur in geometry. For
example, a rotation through 120° about the centre of an equi-
lateral triangle ABC maps the vertices onto each other in the
following way: A' = B, B' = C, C = A (see Figure 3). Here
we have a mapping of the finite set {A,B,C} onto itself. One-
one mappings of a finite set onto itself are called permutations.
In this monograph we shall consider almost exclusively one-
one mappings of the set of all points in the plane onto itself.
We shall use the word transformation to describe this type of
mapping.
The transformations of the plane onto itself which are con-
sidered in elementary geometry can all be based on construc-
tions with compass and ruler. This course develops some topics
of plane geometry using these transformations as a basis.
REFLECTION IN A LINE
Let s be some selected line in the plane. Let us now imagine that
ail the points of the plane are traced onto a sheet of transparent
paper. If we turn this sheet over in such a way that every
point originally on ,v coincides with its original position, then
we obtain a correspondence between points of the plane by
transferring the points from the tracing paper back onto the
plane. We call this mapping of the plane onto itself a reflection
in the line s and we denote it by the symbol M s * We shall call s
the axis of reflection.
Reflection in a line is a transformation having the following
obvious properties on which we shall base our further investiga-
tions.
1. To every line s there corresponds a reflection A/,. If A' is the
image of A under M„ then A is the image of A' (see Figure
4(a)). We write
M,{A) - A', MIA') = A.
2. The points of s remain in the same position; they are the
fixed points of the mapping M s . We say that s itself is a fixed
line of the mapping. If A is not on s, then ^4' lies on the
opposite side of s. The line AA' is also a fixed line ; it is,
however, not pointwise fixed like s (see Figure 4(a)).
3. The reflection M, preserves straight lines, that is, a straight
line, g, has as its image another straight line, g' (see Figure
4(a)).
* We here follow the current British notation; that is, we shall denote
rotations by R and reflections by M (for mirror).
20
REFLECTION IN A LINE
21
.4 = 0'
,F=F< A'=B,
S\
s A>
o—
(Q)
Fig 4
(b)
4. The reflection M s leaves distances and angles invariant. A
closed polygon and its image, however, have opposite
orientation (see Figure 4(b)).
A particular consequence of 4 is that the angles and 0'
shown in Figure 4(a) are equal. This implies that AA' and s are
perpendicular lines.
Together with the relation AF = A'F this gives a simple
construction for obtaining images under the reflection M 3 : the
line segment joining corresponding points A and A' is perpen-
dicular to s and is bisected by it.
5. If two different points A and A' are given arbitrarily, then
there exists exactly one reflection M s such that M S (A) = A'
and M ,(A') = A, The axis s is the mediator of the segment
AA'.
6. Given two rays FA and FB issuing from a point F, then there
exists exactly one reflection M s interchanging the two rays.
The axis s is the bisector of the angle AFB.
Problem 1. Given s, a point A, A' = M S (A% and a point X,
construct the image point A" using only a ruler (that is, only
straight lines may be drawn).
Hint: remember that if B = A', then B' = A. Every pair of
corresponding points, therefore, defines a second pair.
22
TRANSFORMATION GEOMETRY
Problem 2. Given are a line s and two points A and B on the
same side of 5. Find the point X on 5 for which the length of the
path composed of the two line segments AX and XB is a
minimum.
Imagine the segment XB reflected in s. Its length is not
changed, that is, we have
AX+XB = AX+XB'.
We can immediately find the minimum of AX+XB'; it is
given by the segment AB'. The two parts AX and XB of the
minimal path lie on symmetric lines g and g' (see Figure 5). The
point X is characterized by the fact that the two lines intersect
s in three equal angles.
According to Fermat's principle, light follows the path for
which the time of travel is a minimum. In a homogeneous
medium minimum time of travel is equivalent to minimum
distance travelled. Reflection of light in a plane mirror, there-
fore, leads to Problem 2. Figure 5 provides a basis for the
solution by construction of many problems on reflection.
Problem 3. A ray of light issuing from a point A is to be
reflected in two lines St and s 2 in such a way that it finally
passes through a given point B (see Figure 6).
The solution can be developed from that given for Problem
2. Now, however, we must make two reflections.
In the solution shown in Figure 6, A has first been reflected in
REFLECTION IN A LINE
23
Fig 7
s t and its image A i mapped onto A 12 by reflection in the lines 2 .
Another possible method of solution is shown in Figure 7. Here
B has been mapped onto B 2 by reflection in the fine s 2 . In
both constructions the line segments drawn (namely, A l2 B and
A l B 2 ) have the same length as the path of light from A to B. It
should be mentioned that while the problem always has a
geometrical solution this does not mean that there is always a
sensible physical solution. For example, for a solution to have
any physical meaning the two points of reflection P and Q must
lie above the intersection of the lines s, and s 2 .
We now return to Figure 6. Reflection in s 1 maps g onto g y
which is mapped onto g t2 by reflection in s 2 . We may take g 12
as being obtained directly from g if we interpret the result of
performing the two reflections, one after the other, as a new
transformation. This expresses an essential idea of transforma-
tion geometry : transformations can be combined. The result of
this composition is a new transformation.
In order to clarify this idea we now consider the combination
of two reflections away from our particular construction prob-
lem. We map the points of the plane first by the reflection M a in
a line a, and then map the images obtained by a reflection M b .
The correspondence between a point A and its final image
A l2 = M b (M a (A)) defines a new transformation which we shall
denote by M b M„. Note that this notation indicates the order of
composition as 'first M a , then M b — as is usual with functional
notation we read from right to left. Figure 8 illustrates the
construction of this new mapping.
The process of combining transformations just described is
24
TRANSFORMATION GEOMETRY
Reflection M a Reflection M b
«-M a MJ-
Combined Transformation M a M a
■M b (M e (g)i
not restricted to reflections in lines; it produces from two
arbitrary transformations <& and *P a new transformation NKO.
This symbol means that each point is first mapped by <D, then
its image is mapped by V.
Now we can see that the set of all mappings of the plane onto
itself has a property we have already met in algebra. Combina-
tion of transformations is an operation associating with any
two transformations a new transformation, that is, an object of
the same kind. We find a similar situation, for example, with
the set of all positive numbers if we consider multiplication as
the operation of composition. By stressing these analogies our
geometric considerations will assume an algebraic aspect.
If a reflection M, is performed twice, we obtain the mapping
which leaves all points of the plane in the same position, that is,
every point of the plane is a fixed point. This trivial mapping is
called the identity transformation and it is denoted by the
symbol /. In the set of all transformations, / plays a similar
part to that of the number 1 in the set of numbers mentioned
above ; for every transformation <& we have
REFLECTION IN A LINE
25
Definition. If we reverse the correspondence between points
given by a transformation <5, we obtain the inverse transforma-
tion to <D. This is denoted by <D~ l .
*,
It follows from this definition that, for every transformation
<J»(I> ' = O '<!> = /'.
Definition. Any transformation (other than the identity) which
is the same as its inverse is called an inuolutory mapping or
involution.
From d> _1 = <J> we deduce that an involution satisfies the
equivalent relation
O* = * 2 = /.
Reflections in lines, therefore, are examples of involutions.
Problem 4. Show by means of suitable drawings:
(a) M p Mq is not, in general, the same transformation as
A/,M p , that is, the combination of reflections is non-commuta-
tive;
(b) M p M q = M t Mp if jp and q are orthogonal lines.
Figure 9(a) illustrates the first assertion.
26
TRANSFORMATION GEOMETRY
fcr
J>£
T
dp.
Fig 9
'(b)
So as to prove the second assertion, we show that every point
A is mapped by the combined transformation onto the same
image irrespective of the order in which the reflections are
carried out (see Figure 9(b)).
»/fj — *A X '
MM.
M q M p
M p M q
A,'
From the invariance of distances and angles we deduce
A,0, ^,'arecollinearand OAJ = OA\ ^ ^ , = A , _ j,
A, O, A 2 ' are collinear and OA 2 ' = OAf ' * 2
The mapping M f M q = M q M p is easily constructed if we note
that is the mid-point of the line segment A A'. We call this
transformation reflection in the point O or a half-turn about O
and we denote it by H .
From the relation
H 2 m H H = M p M q M„M p - /
we see that a half-turn is an involution.*
* We have here assumed that the operation of combining transforma-
tions is associative, that is, that {MpM^(MJtf^l = M„((M,M,)M S ). The
reader should check the validity of this assumption for himself.
REFLECTION IN A LINE
27
In connection with this problem we want to mention briefly
the question of parallels. The quadrangle AA 1 A'A 2 is obviously
a rectangle. However, we can only prove this if we use some
axiom which expresses, in some form or other, Euclid's postu-
late on parallels. If we want to base plane geometry on the
study of transformations, the following postulate appears to be
especially suitable.
If three angles in a quadrangle are right-angles, then so is the
fourth angle. (Axiom of the existence of the rectangle.)
From this we deduce that the quadrangle OPAxQ is a
rectangle. The same is true for the three other quadrangles in
corresponding positions. But then we have proved that all the
angles in the quadrangle AA^A'A-l are right-angles.
Problem 5. Show that for every rectangle there exist two
reflections M p and M q which map the rectangle onto itself. The
axes p and q are orthogonal.
Definition. If a reflection M p maps a figure onto itself, then we
say that p is an axis of symmetry of the figure. If the figure is
mapped onto itself by a half-turn H , then we say that O is a
centre of symmetry of the figure.
Figure 10 shows two reflections which map the square ABCD
onto itself, the axes of symmetry being s t and s 2 . The square
has four axes of symmetry; a regular H-sided polygon is easily
shown to have n axes of symmetry. For a circle every diameter
is an axis of symmetry.
Wallpaper patterns, especially friezes, supply us with many
simple examples of figures with infinitely many axes or centres
D*C'
9—
a —
C=D'
* >
B=A>
28 TRANSFORMATION GEOMETRY
of symmetry. (We must here assume that the pattern is extended
indefinitely.)
Definition. Two lines g and h are called parallel if both are
orthogonal to a line a.
It is easy to deduce from this that parallel lines are also
characterized by the equality of a pair of alternating angles
with a line a.
Problem 6. Deduce from the definition that:
/■|gandg||/j *>f\\h.
T
Problem 7. We are given two parallel lines p and q and two
points A and B between them. We wish to construct the path of
a ray of light issuing from A which passes through B after being
reflected twice in both p and q, the first reflection being in p.
Let g be the line along which the light begins to travel. Then
the subsequent parts of the path lie on the image lines of g under
the mappings M p , M q M p , M p M q M p and M q M„M q M p . The
problem can be solved by constructing the images of A under
these transformations, namely A u A 2 , A 3 and A'. The line
through A' and B is the required Imeg' and once this has been
drawn the remaining parts of the path can be found easily.
Note that we could save a little space by making use of the fact
that M„ and M q are involutions. For
M q M p M„M p {g) m g'
REFLECTION IN A LINE 29
and, therefore,
M p M q {M q M p M q M p {g)) = M p M q (g').
But, using the involution property,
M,M q M q M p M q M p (g) = M q M p (g).
That is, we can obtain g 2 by considering the line joining
A 2 = M q M p (A) to B 2 = M p M q (B). This obviates the need to
construct A'.
Problem 8. Solve Problem 7 if only three reflections are de-
manded (for example, one in p and two in q).
Problem 9. A rectangle has width p and length 2p. The sides of
the rectangle lie on the lines a, b, c and d, and its centre is M.
Construct the hexagon MA { BCDA 2 with A x and A z on a, B on
b. C on c and D on d, which has the smallest perimeter subject
to these conditions. Express this minimal perimeter in terms of
P-
We can solve the problem by considering the image of M
under the transformation M e MiM c M b M a .
However, we prefer to split this chain into the products
M c M b M B and M a M d : this leads us to the two points M 3
= M c M b M a {M) and M 2 ' = M 4 M a {M) (cf. Problem 7). The
30
TRANSFORMATION GEOMETRY
minimal perimeter is equal to the length of M 3 M 2 '. We find
that it is 5p,
If we consider the symmetry of this problem, we can see
immediately that C is the centre of a side of the rectangle. If C
is known the construction can be considerably simplified.
In the solutions of Problems 3, 7, 8 and 9 we have met
products of a finite number of reflections. These are transforma-
tions leaving distances and angles unchanged. Hence, if a figure
is mapped by a product of a finite number of reflections, then
its image is congruent to the original.
Definition. Finite products of reflections are called congruences
or isometries.
Fig 73
Figure 1 3 represents a product of three reflections
We can see from the figure that the inverse transformation is
*~ l = M p M q M r .
given by
Theorem 1. We can find the inverse transformation to a finite
product of reflections by reversing the order of composition of
the reflections.
RM-LIXTIOX IN A LINi;
31
After these remarks we turn to some further types of prob-
lems whose solutions can be found by means of reflections.
Problem 10. A line s and two circles T u T 2 are given. Construct
squares with two opposite vertices on s and with one of the
remaining vertices on each of the circles T, and F 2 .
Rg-H
Assume that A and C lie on s. The vertices B and D are then
symmetrically placed with respect to s t that is, M S (B) = D.
Imagine all the squares that could be drawn with X and Zona
and Y on T t . The locus of the vertices Y' would then be the
image circle F/ (= MJtTx)). The vertex of the square to be
constructed must, therefore, be a point of intersection of F t '
and F 2 .
This solution is based on the reflection of a locus of points.
This is a new way of using mappings to help us to solve con-
struction problems.
Problem 11. A line s and two circles r 1( T 2 on the same side of s
are given. Construct a point X on s with the property that
tangents from X to the two circles make equal angles with s.
32
TRANSFORMATION GEOMETRY
The condition means that the two tangents through X are
symmetric with respect to the axis s, that is, M s maps one onto
the other. Therefore, X must be the intersection of s with a
common tangent to the circles r t and T 2 '. In general, therefore,
the problem has four solutions.
Problem 12. A line g and two points A, B on the same side of g
are given. Find a point X on g such that the angle between XB
and g is twice as large as that between XA and g.
TRANSLATIONS
Fig 16
We now consider the product of two reflections in parallel
axes p and q (see Figure 16).
M q M p maps A onto the point A' on the perpendicular to p
and q through A; the distance AA' being twice the distance
between p and?. If we consider this new transformation without
reference to M p and M q , we can characterize it as follows :
The Hues joining corresponding points P and P' are parallel.
All points are moved the same distance in the same direction.
Definition. A mapping satisfying the above conditions is called
a translation.
A translation is obviously determined if one arbitrary pair
P, P'— consisting of a point and its image— are given. This pair
determines a direction and a distance— two quantities which
can be represented geometrically by an arrow leading from P to
P'. The arrows associated with different pairs of corresponding
c 33
34
TRANSFORMATION GEOMETRY
Fig 17
points are all equivalent since each of them determines the same
translation.
Definition. A class of equivalent arrows is associated with every
translation. The equivalence class of arrows is called a vector.
Every one of the arrows serves to represent the vector.
We denote vectors in the text by small letters printed in bold
type, for example, a, b, «, v, and in the figures by the symbol —
placed below the letters.
If a translation is determined by a vector v, we find the images
of the points A, B, C, . . . by attaching the vector v to each of the
points in turn (see Figure 17).
We now return to the generation of translations by composi-
tion of reflections (Figure 16). The translation resulting from
the combination of M p and M q has a very simple relation to the
lines/) and q: the translation vector, v, is twice as long as, and
in the same direction as, a, the vector perpendicular top which
runs from a point on p to a point on q. Here we have to note the
order of composition; for the mapping M q M p the vector a
points from p to q. We express the relation between v and a by
the equation v = 2a.
We now consider a translation T with vector v. If p and q
are two parallel lines such that the vector a, which is perpendi-
cular to p and runs from p to q, satisfies the equation a = ^r,
then T = M q M p . Since there are infinitely many pairs of parallel
lines which satisfy these conditions we have the following
theorem.
TRANSLATIONS
35
Theorem 2. A translation T with vector v can be represented in
infinitely many ways as a product of two reflections. The axes
of the two reflections are parallel, are orthogonal to v, and are
a distance apart equal to half the length of p.
We shall use this theorem to help us to analyse isometries.
It is obvious that translations, being products of reflections,
are isometries. We deduce the following properties.
1. Translations are one-one mappings.
2. Translations are line-preserving transformations; moreover,
the image g' of a line g is always parallel to g. (See Figure
18.)
3. Translations are direct isometries, that is, congruences which
preserve orientation.
Fig 19
FigtB
The inverse transformation to a translation Twith vector v is
again a translation. Its vector has the same length as v, but
opposite direction; it is denoted by — v (see Figure 19).
Before investigating translations further we shall consider
two elementary constructions based on translation.
Problem 13. Two lines g and h are given. Construct a line x
making an angle of 60° with g and such that its points of inter-
section with g and h are a distance d apart.
36
TRANSFORMATION GEOMETRY
Fig 20
The angle of 60° with g and the length d determine a vector v
(see Figure 20). A translation with vector v maps g onto g'.
The required line x passes through the intersection of g' and It
and is parallel to v.
The problem has four solutions; it is easy to see that, apart
from v, the vectors w, - v and -w will all lead to lines with the
necessary properties.
Problem 14. Given are two circles T, and T 2 and a \ineg. Find
a point A on Tj and a point B on T 2 such that A and B are a
distance </ apart and A3 is parallel to g.
Here a solution is obtained by mapping one of the circles by
means of a translation with a vector a which is parallel to g and
has length d.
COMBINATION OF TRANSLATIONS
When we consider the product of translations the half-turn,
which was defined on p. 26, will prove to be an extremely useful
aid and so we shall now consider this particular congruence in
more detail.
First we show that a translation can be represented as the
product of two half-turns. For this purpose we split the trans-
lation T into two reflections M p and M q (see Figure 21). If s is
perpendicular to p and q, then
T - M q M p = M q M,M s M p = (M t M s )(M s M P )
— fJ g H f ,
using the fact that the product of reflections in orthogonal lines
TRANSLATIONS
37
Fig 21
Fig 22
is a half-turn about their point of intersection.
We immediately deduce:
Theorem 3. A translation T with vector v can be represented in
infinitely many ways as a product of two half-turns. The
centres of the half-turns F and G have to be chosen in such a
way that FG = $v.
Problem 15. Show that the product of three half-turns is again a
half-turn.
Let the product be £1 = H„H G H F .
We solve the problem by considering the translation
H G H e = T, which has the vector v = 2FG. Choose the point K
such that KH = FG (see Figure 22). Then we have
« - H H {H G H F ) m HJPM = H K ,
which proves our assertion.
Opposite sides of the quadrilateral FGHK are parallel and of
equal length. This can be shown by multiplying
H C H F = H,[H K
by H F on the right and by H n on the left. We then obtain
ffii^G^F^r ~ HnHtiHicHf <*■ H n H c = H K H F .
But this implies GH = FK. Hence the points F,G,H,K are the
vertices of a parallelogram.
Remembering that a half-turn is an involution, we deduce the
relation
« = rif[Jtj G ii F HijM G H F = /.
38 TRANSFORMATION GEOMETRY
Problem 16. Find the image of the point A under ft 2 by finding
successive images under half-turns about F,G,H,F f G,H- Of
what geometric figure do these image points form the vertices ?
We note that
Theorem 4. In a product of three half-turns the order of the
factors may be reversed.
Problem 17. Let T be a translation with vector v and let H F be
a half-turn about f. Show that TH F and H f Tzltc also half-turns
and construct their centres from v and F.
We now consider two translations 7\ and T 2 with vectors a
and b respectively. We split each translation into two half-
turns:
Fig 23
Combining T t and T 2 we obtain the mapping
r 2 7\ = H B H G H C H P = H H H Fy
which is clearly again a translation, being the product of two
half-turns. On the other hand
T X T 2 = H G H F HffH G = {H g H f Hj^Hq
= {HiiH F H G )H G = H B H F .
Here we have used Theorem 4 which said that the order of
factors in a product of three half-turns could be reversed.
Comparing the two representations for T 2 Ti and T t T z we
find
T t T 2 = T 2 Ti = T.
TRANSLATIONS
39
Fig 2'i
£*£*&
Theorem 5. The product of two translations is a translation.
Moreover, the order of composition does not matter, that is,
combination of translations is commutative.
We associate with the new translation T the vector c. It can
be obtained by considering the map of a single point A. The
image of A under 7", is A* and T 2 maps A* onto A'. The vector
c is, therefore, represented by an arrow from A to A'. We call
c the sum of the two vectors a and b, and write
c = a+b.
This associates with the operation of combining translations
an operation defined for vectors which is called vector addition.
We obtain the vector sum of a and b by attaching the vector
b to the 'point' of the vector a. The vector sum is then the vector
which starts at the starting point of a and ends at the end-point
of b (see Figure 24). This construction appears to depend
upon the order in which the addition takes place, but this does
not influence the result since
T 2 Ti = T 1 T 2 *>a+b = b+a (see Figure 25).
40
TRANSFORMATION GEOMETRY
b
Fig 25
Fig 26
Translations and vectors provide us with two ways of
expressing the same geometrical facts. The product T Z T % in the
language of translations corresponds to the vector sum a+b.
If we perform the same translation twice, we describe this by
T = TiTi = T^o-c = a + a = la.
The vector c has the same direction as a, but is twice as long
(see Figure 26).
If 7*i is performed n times, we have
T = T t * o e = a+a+ ... +a = na.
n times
Here we have introduced the symbol na for the vector which
has the same direction as a, but is n times as long.
For later applications we shall need the following generaliza-
tion.
Definition. Let / be a real number. We denote by Xa the vector
with a length equal to the length of a multiplied by \X\, and
which has the same or the opposite direction as a according to
whether /. is positive or negative.*
This definition includes the previously defined vector —a;
this is the vector Xa for X = — 1.
Certain geometric properties lead to further algebraic
operations for vectors. The theory of these operations is called
vector algebra. We are here only interested in vector addition
•The definition uses implicitly the following principle of continuity:
if a line segment is given on a line, then every other segment on the line can
be characterized by a real number: its measure in relation to the given line
segment.
TRANSLATIONS
41
and in multiplication of vectors by a number. A further study
of vector algebra is not, therefore, necessary for our purposes.
Vectors are of great importance in physics. Most physical
quantities are vectors or similar to vectors. We mention as
examples forces, velocities, accelerations, electric and magnetic
fields.
GROUPS
We now consider the set £ of all translations. Included in
this set is the identity translation which is interpreted as the
translation with a vector of length zero.* We found that when
two translations were combined the resulting transformation
was again a translation (Theorem 5). Hence, when two ele-
ments of % are combined they always yield another element of
X, The operation of composition does not take us outside the set
X It has further been shown that associated with every transla-
tion there is an inverse transformation which is also a transla-
tion. Every element ofX has an inverse which belongs to X. Such
sets with an operation defined upon them having the above
properties are extremely important in modern mathematics.
They are known as groups. So as to enable us to develop this
idea we now give a precise definition of a group.
Definition. A set <S of objects (elements) A,B,C,
group if the following postulates are satisfied.
is called a
I. An operation is defined on © which associates with every
ordered pair A, B of elements of ® a new element C which
also belongs to (5, C is called the product of A and B and we
write
C = AoB.
II. There exists an element E of t5 satisfying the relation
AoE = EoA = A
for all elements A of ®,
£ is called the unit, or neutral, element of the group.
• The direction of this vector is irrelevant. In vector algebra it is known
as the zero vector.
42
TRANSFORMATION GEOMLTRY
III. For every element A of <5 there exists an inverse element,
denoted by A~ l , which satisfies
AoA~ l m A~ l oA = E.
IV. The operation is associative, that is,
(AoB)oC m Ao(BoC).
Thus translations form a group when we take the operation of
composition to be the result of performing the two mappings
one after the other. The unit element is the identity translation.
The inverse element corresponding to a translation with vector
v is the translation with vector — v. Postulate IV is automatically
satisfied by the operation of combining transformations.
. t
Ai>
~H
^3
Fig 2?
Further examples of groups of transformations may be
obtained, for example, by considering only those translations
which map a certain figure onto itself. Let us ask, for instance,
which translations map the frieze pattern shown in Figure 27
onto itself. (We imagine that the pattern has been extended
indefinitely in both directions.) One of them is certainly the
translation with the vector v. But the translations with the
vectors 2v, 3p, — p, — 2v, , . . also leave the figure unchanged.
The figure, in fact, is left unchanged by any translation with a
vector of the form Xv, where X is an arbitrary integer. It is easy
to verify that the set of all such translations forms a group. We
obtain the group of all translations which map the given linear
pattern onto itself.
The parquet pattern shown in Figure 28 is also left invariant
under certain translations provided we think of the pattern as
being continued indefinitely. If two such translations are
combined, then we obtain another mapping of the pattern onto
TRANSLATIONS
43
itself. The translations with vectors v lt v 2f P3, v 4 are examples
of transformations which map the pattern onto itself. It is easily
seen that all the translations with this property have vectors of
the form
V sa Ap,+jif 2j
where X and ft are integers, for example, p 3 = 2v l + v 1 ,
p 4 = p 1 +2p 2 . These mappings again form a group; we call it
trie group of translations of the given plane pattern.
The idea of a group is not restricted to transformations. The
definition of a group applies to other sets and operations. The
study of such systems has developed into a special branch of
mathematics— group theory. This theory provides a common
framework for statements and methods concerning the most
diverse fields of mathematics, provided that they are based on
the same logical structure. Here one encounters mathematical
thinking in its purest form.
In connection with the two groups of translations of patterns
we should like to remark that groups of mappings of figures are
sometimes used to characterize works of art, patterns and types
of crystals.*
•See, for example, Coxeter, Introduction to Geometry, pp. 35, 278-9;
Terpstra, Some notes on the Mathematical Background of Repetitive
Patterns; Weyl, Symmetry.
44
TRANSFORMATION GEOMETRY
The group of translations which leave the linear pattern of
Figure 27 invariant contains only transformations which are
also elements of the group of all translations. Here we have a
subset of the full group of translations which is itself a group.
One talks in this context of a subgroup.
Definition. A subset S of a group © is called a subgroup of (5 if
the elements of § themselves form a group under the operation
defined on (5.
The group of translations which leave the plane pattern of
Figure 28 invariant is also a subgroup of the group of all
translations.
Problem 18. A square A BCD is given. Let a be the vector with
starting point D and end-point B, and let b be the vector from
D to C. Find the image of the square under the translation with
the vector 2a +36.
Problem 19. In a square ABCD we define vectors u = AB
and v = AD. What is the figure formed by the images of this
square iT it is mapped by all the translations having vectors
-?.B+;u* where A and /* are integers?
FURTHER EXAMPLES OF GROUPS
1. The group of integers under the operation of ordinary
addition. The unit element is the number and the inverse
element to the number n is —n.
2. The group of even integers under the operation of addition.
This is a subgroup of the previous example.
3. The group of rational numbers under the operation of
addition. This group contains the two previous examples as
subgroups.
4. The group of non-zero rational numbers under the opera-
lion of ordinary multiplication. Here the unit element is the
number 1 and the inverse clement to n is l/«.
5. The group of positive rational numbers under the operation
of multiplication. This is a subgroup of the group of
Example 4.
TRANSLATIONS
45
Let E be the symbol for an even number and O the symbol
for an odd number. We take ordinary addition of numbers
to be our operation of composition. Then we obtain the
following rules of addition :
E+E = E (even + even = even)
£+ = (even + odd = odd, independent of order)
O + O = E (odd + odd = even).
This defines the operation of addition on the set {0,E} and
we can easily verify that these two elements form a group
under the operation + . When a group contains only a finite
number of elements, then we can describe it by means of a
group table such as that below.
Second
element
+ ! E O
First E
clement
E
O E
1. We divide all the integers into three classes according to the
remainder left when they arc divided by 3. We use the symbol
R for all integers divisible by 3, R^ for those which leave
the remainder 1 and R 2 f° r those integers which leave the
remainder 2. There are no other integers. If, once more, we
use addition as our operation, we obtain the following
rules:
Rq+Rq = Rq,
R l + R i = R 2 ,
Ro + R^ = Ri,
R 1 + R z m R ,
Rq + R 2 = ^2'
R 2 + R 2 — Ri-
We can easily check that this system forms a group with
three elements. The group table has the following form :
+
R Q R t R 2
Rq
Rq Ri R 2
*i
/?! R 2 Rq
R 2
R 2 Rq Ri
46 TRANSFORMATION GEOMETRY
The three symbols R , Ry and R 2 represent the residue
classes modulo 3 ; this group is, therefore, called the group of
residue classes modulo 3. Here R is the unit element: the
inverse element to S u for example, is R 2 .
Problem 20. Check whether or not the following sets form
groups under the given operations.
(a) The set (5 of ali positive fractions of the form l//i, under
multiplication.
(b) The set SR of all residue classes modulo 5 other than the
zero class (that is, the set 91 = {R u R 2 , R 3 , R*}), under
multiplication.
(c) The set SR of all residue classes modulo 6 other than the
zero class, tinder multiplication.
(d) The set K of all integers, under subtraction.
(e) The set $ = {a+by/2:a,b rational; a 1 +b z ^0}, under
multiplication.
{/)The set of all translations with vectors of the form Xa+pb
where a and b are given vectors and X, ji are irrational num-
bers, under the operation of combination of transforma-
tions.
We conclude this first introduction to groups with an example
of a non-associadve operation. Let ty be the set of all points in
the plane. With each pair of points A and B in ^3 we associate
the point C which is the mid-point of the line segment AB. The
reader should verify that the operation so defined is not asso-
ciative. This example shows that one must not always assume
the associativity of an operation.
ROTATIONS
Fig 29
We now consider the product of two reflections M p and M q
whose axes, p and q, intersect in the point O. The transforma-
tion R = M q M p is a congruence which preserves orientation.
Such congruences are known as direct isometries. The position
of corresponding points A and A' = R(A) is such that
OA = OA' and the angle
= LAO A'
has a fixed value. We have in fact
L.AOA' = 2« 1 +2« 2 = 2<t»,
where <b is the angle between the axes of reflection p and q
(see Figure 29).
The mapping R is called a rotation about O through the angle
= 2<j>. 6 is known as the angle of rotation and O is the centre
of rotation.
Problem 21. Verify the relation /_AOA' = 2<f> for different
positions of A.
47
48
TRANSFORMATION GEOMETRY
A rotation R is obviously determined once the centre and
angle and direction of rotation are known. Angle and direction
of rotation can be considered together as one quantity known
as a directed angle. In this respect it is usual to consider angles
to be positive if the direction of rotation is anticlockwise. Our
statement concerning the product of M p and M q can now be
written:
Theorem 6. If p and q are two lines which intersect at O, then
M q M p is a rotation about and the angle of rotation is twice
the directed angle between p and q.
At the same time we can deduce the following theorem :
Theorem 7. A given rotation R about a point O and through a
directed angle can be expressed in infinitely many ways as the
product of two reflections. The axes of reflection p and q can
be any pair of lines through O such that <f>, the directed angle
between p and q, satisfies the relation
Using Theorem 7 all rotations can be reduced to a product of
reflections in exactly the same way as we found to be the case
for translations. This will prove to be of value when we come to
analyse isometrics.
We can immediately deduce the following properties of
rotations.
^
ROTATIONS
49
Fig 30
1. A rotation is a one-one mapping; the inverse mapping to
R(0,8) is the rotation ^"'(0,-0).
2. Rotations are line-preserving transformations. The angle
between a line g and its image g' is equal to the angle of
rotation.
This property is easily verified from Figure 30. The line h
is the perpendicular to g through O and h' is its image. The
angle between g and g' is y = 0.
3. A rotation has, in general, exactly one fixed point, namely
the centre of rotation 0. The exceptional cases arise when 8
is a multiple of 360° in which case every point is a fixed point,
that is, we have the identity transformation.
We shall now familiarize ourselves with this new type of
transformation by making use of rotations to solve some
construction problems.
Problem 22. Rotate a square A BCD about a given point
through an angle 8 = —60°.
Problem 23. A point A and two lines b and d are given. Con-
struct a square ABCD such that B lies on b and D on d*
We begin the solution of this problem by ignoring the
condition on D. The remaining conditions determine an easily
identifiable family of squares. If we draw the square with
vertex X on b, then the opposite vertex X' is the image of X
under a rotation about A through an angle 8 = 90° (see Figure
3 1). If X moves on b, then X' moves on the image of b under
this rotation, that is, on the line b'.
The problem has two solutions, since the rotation through
8 = - 90° will also provide a solution.
The solution to Problem 23 depended on the mapping of a
geometric locus by a rotation. This same idea can be used to
solve Problems 24 to 30.
* We assume that the vertices of the square are A, B, C, O in that order,
cf. Problem 95.
50
TRANSFORMATION GEOMETRY
Fig 31
Problem 24. Construct an equilateral triangle which has one
vertex at A and one vertex on each of the given lines b and c.
Problem 25. A point A and two lines b and c are given. Con-
struct an isosceles triangle ABC such that B lies on b, C lies on
c and I ABC = LACB = 75°.
Problem 26. Given are a point A and two concentric circles
r, and r 2 . Construct an equilateral triangle with one vertex at
A, the vertex B on T x and the vertex C on T 2 .
Problem 27, Given are a point P and two circles T ( and r 2 .
Find a point „4 on T l and a point 5 on T 2 such that /> is the
mid-point of AB.
Problem 28. Given are two circles Tj and T 2 intersecting in the
points P and Q. Construct a line through P which intersects
the two circles in chords of equal length.
For the solution of Problems 27 and 28 we use rotations
through 180°, that is, half-turns.
ROTATIONS 51
Problem 29. Given are three points A, B and C. Construct a
square with centre A such that two adjoining sides (or their
extensions) pass through B and C respectively.
Problem 30. P and Q are two points inside a circle F. Construct
two chords of T of equal length, which intersect at an angle of
45° and such that one passes through P and the other through
Q-
We now turn to another type of problem which can be solved
by means of rotations. Now, however, the angle of rotation is
not fixed ; it depends upon how we draw the initial figure.
Problem 31. A circle T and a point P are given. Construct a
second circle with a given radius r which passes through P and
which has a common chord with T of length s.
We first ignore the condition that the circle should pass
through P. Then there is a family of circles satisfying the
remaining conditions and we can easily construct one of them;
the circle with centre M (see Figure 32) is a representative of
this family. To obtain a solution to the problem this circle must
be rotated about O until it passes through P.
Problem 32. Given are a circle V and a line g. Construct a
triangle with given sides which has the vertices A and Hour
and the vertex C on g.
52
TRANSFORMATION GEOMETRY
Fig 33
If we ignore the last condition we obtain two families of
triangles, one for each orientation of the triangle. In Figure 33
we have drawn one representative of each of the two families.
A suitable rotation about O will bring C x or C 2 onto the line 3.
Problem 33. Two concentric circles Fj and T z , and a line g are
given. Construct an equilateral triangle having sides of given
length, d, and one vertex on each of T lt T 2 and g.
We now continue our study of the theory of rotations by
considering yet another type of problem.
ROTATIONS
53
Problem 34. AB and A'B' are two given directed line segments
having the same length but not parallel. Show that there always
exists a rotation R which maps A onto A' and B onto B'.
The problem is solved if we can find two reflections M p and
M„ such that M M p (A) = A', M q M„(B) = B'.
We take p to be the mediator of AA' and let B* denote the
image of B under M p . If we now take q to be the mediator of
B*B', then M p and M 4 are the two reflections which satisfy our
requirements. The centre of rotation (the point of intersec-
tion of p and q) can also be found by a construction based on
symmetry: O is the point of intersection of the mediators of
A A' and BB' (see Figure 34). From this we can see that there is
essentially only one rotation giving the required mapping (that
is, there is only one rotation if we restrict the angle of rotation
to lying between and 360°).
Problem 35. What happens to the construction above if the
line segments AB and A'B' are
(a) symmetric with respect to a line,
(b) parallel but in an opposite sense?
Under the assumptions about AB and A'B' which are listed
in Problem 34 we can always find a rotation having the required
properties. It seems reasonable, therefore, to see what happens
in the exceptional case when the two directed line segments are
parallel.
Assuming that AB and A'B' are parallel directed line seg-
ments having equal lengths, then our construction leads to
parallel axes of reflection. Hence, the transformation that
Fig 35
54
TRANSFORMATION GEOMETRY
maps AB onto A'B' is a translation. Translations can be con-
sidered as special cases of rotations, namely as rotations about
a centre at an infinite distance through a zero angle.
Fig 36
Problem 36. A rectangular table top of length 2 yd. and of
width 1 yd. is to be fixed to the supporting frame in such a way
that, by turning about an axis perpendicular to its plane, it can
take the two positions indicated in Figure 36. Is this possible,
and if so, about which point must the table top rotate ?
Using the labelling of vertices indicated in Figure 36, the two
rectangles ABCD and A'B'C'D' are congruent and have the
same orientation. Since corresponding sides are not parallel,
there must be a rotation which maps one onto the other. The
centre of rotation is the point of intersection of the mediators of
line segments connecting corresponding points. If we take
A, A' and M,M' as the pairs of corresponding points, we
obtain mediators which are orthogonal and, moreover, parallel
to the sides of the rectangle. The coordinates of O with respect
to ABCD are then easily calculated.
It should be noted that there are two essentially different ways
of associating the vertices of the two rectangles. Consequently
the problem has two solutions. In Figure 36 the angle of rotation
is Q m +90°; for the other solution the angle is = -90°.
ROTATIONS 55
COMBINATION OF ROTATIONS
We now consider the transformation resulting from the
composition of two rotations. We shall see that, in general, we
obtain another rotation and this will lead us to a new group of
transformations.
We start with:
Problem 37. Map the square ABCD by the rotation
Ri(Oy,0i = 60°) and then map its image by the rotation
R 2 (O 2 ,0 2 = 90°). Show that the resultant mapping is again a
rotation.
Figure 37 shows the various steps of the construction. The
mapping R 2 R t is obviously an isometry which preserves
orientation, and it must again be a rotation. For, according to
Problem 34, there is exactly one rotation which maps A D onto
A'D'. This mapping, however, maps ABCD onto A'B'C'D'.
The centre of rotation O and the angle of rotation are easily
constructed from AB and A'B'.
Fig 37
56
TRANSFORMATION GEOMETRY
/
/
Fig 3d
We analyse the product of two rotations further by consider-
ing a suitable decomposition of J?, and R 2 into reflections. Let
s be the line through the two centres of rotation, O t and 2 . By
Theorem 7, we can decompose J?! and J? 2 in infinitely many
ways into two reflections. We now select two lines p and q such
that
K, = M S M P , R 2 = Af,Af a .
Then
R = R 2 R V = M q M s M s M P = M,M r
The fixed point of R is the point of intersection of/? and q :
the angle of rotation is twice the directed angle between p and q.
From Figure 38 we immediately deduce the relation
= 0i+0 2 .
Problem 38. Find, using reflections, the product of two rotations
R t and R 2 when the angles of rotation 8 t and 9 2 have different
signs.
If we take into account the signs of the angles we once more
find that
8 = 8 t + 9 2 (see Figure 39).
Fig 39
ROTATIONS
57
Theorem 8. When rotations are combined, the angles of rotation
are added algebraically.
A"fes: —
h^lX^
Fig 40
ft =-30
$2= +30°
Fig. 41
If the two angles of rotation differ only in sign, that is,
8y+8 2 = 0, then every line g is mapped onto a parallel line
g' (see Figure 40).
Figure 41 illustrates how this case is analysed using reflec-
tions. We can see immediately that R 2 Ri = M t M p is a transla-
tion.
58 TRANSFORMATION GEOMETRY
Problem 39. Given are a translation with vector v and a rotation
about the point O through an angle = 60°. The product of
the two mappings, in the given order, is again a rotation through
6 = 60°. Construct the centre of this rotation.
Theorem 9. Rotations and translations together form a group S
under the operation of combination of transformations.
To prove this theorem we have to verify that all the group
postulates are satisfied. This is easily done, using the results
already obtained, and we leave it to the reader.
If two rotations R t and R 2 are performed one after the other,
the final position of a given figure will, as a rule, depend upon
the order in which the rotations are performed. Rotations are
not, in general, commutative. This is best shown by an example.
Let us take R t to be a rotation about the point 0, through an
angle Q t = — 60° and let R 2 be a rotation about 2 through
0, « -90°.
Consider the effect of the two mappings, one after the other,
on a simple figure such as the line segment AB joining the two
centres of rotation. The construction shows that the position
of the final image of AB depends upon the order in which the
two rotations are performed (see Figure 42). The mapping
R 2 Ri (that is, first R t then R 2 ) maps AB onto A V 'B X '\ R X R 2
maps AB onto A 2 'B 2 .
The angle of rotation in both cases is t +0 2 . Consequently
the two image line segments are parallel.
The group of all translations and rotations, therefore, differs
essentially from the group of all translations since the product of
two of the group elements now depends upon the order in
which the two elements are written. We say that the group is
non-commutative. We note, however, that there are certain pairs
of mappings in the group which do commute. For example, the
product of two rotations about the same point is independent of
the order in which the rotations are performed. However, the
commutative law no longer holds in general.
Problem 40. R t and R 2 are two rotations. Construct the centres
of R 2 Ri and R l R 2 using suitable decompositions of R t and R 2
into products of reflections.
"
ROTATIONS
59
B,=B> 2
Fig 42
Fig. 43
Let s be the line between the two centres, we can then use
the following decomposition which is indicated in Figure 43:
R t m M,M p = M,.M„ R 2 = M € M S = Af.jtf,.
We obtain
R 2 Rl = M q M p , R t R 2 m M p .M q ..
The fixed points of the two rotations R 2 R X and RiR 2 are,
therefore, symmetrically placed with respect to the line s.
Problem 41. Given are two rotations R x and R 2 . Find those
points which are mapped onto the same image point under i?,
and R 2 .
Hint: a point with this property is a fixed point under R 2 ~ 1 R 1 .
Problem 42. If i? t and R 2 are two arbitrary rotations, show that
the centres of R lt R 2 R t and R 2 ~ l Ri are collinear.
Problem 43. A, B and C are the vertices of a triangle (described
in a clockwise direction), and a, 0, y are the angles at A, B and
C respectively. Show that the three rotations Rt(A, 2a),
R 2 {B t 20), R 3 (C t 2y) satisfy the relation
R$R 2 R) — I,
60
TRANSFORMATION GEOMETRY
Problem 44. Give pairs of elements of the group S which
commute with each other.
Problem 45. Given are four points A, B, Cand D. Find a square
such that each side (or its extension) passes through one of the
given points.
We analyse this problem in the following way (see Figure 44).
A rotation R about M, the centre of the square, through an
angle = 90 s maps the square onto itself. In particular, the
strip between a and c is mapped onto the strip between b and d.
The vector u (from A to C) is mapped onto a vector w which
has the same length as u and is at right-angles to u. The rotation
cannot, of course, be realized immediately since M, the centre
of rotation, is still unknown. However, let us consider the
product of this rotation and a translation T with vector t.
This maps C onto C* = D and maps A' onto A*. The vector
h> is not changed since vectors are invariant under translations —
it is only moved to another position.
The mapping TR can now be constructed. It maps a onto the
Fig. 44
ROTATIONS
61
Fig. 45
vector w which has the above mentioned properties and, more-
over, has its end-point at D. We can now construct this vector
without any difficulty. But once we have constructed w we have
found A*, that is, a second point on the side of the square
through B. This enables us to draw the side b and the remaining
sides of the square can then be readily constructed.
We leave it to the reader to check that this problem has six
different solutions. Figure 45 shows how the square is con-
structed for a special disposition of the points A, B, C, D.
Problem 46. Given are three points A, B, C. Construct a square
which has A as a vertex and such that one of its two sides not
passing through A passes through B and the other through C.
This is a special case of Problem 45. Here two of the given
points coincide.
In the solution of Problem 45 we used a rotation TR which
mapped the strip between the lines a and c onto the strip
bounded by b and d. In particular, a was mapped onto b and c
onto d. There are infinitely many rotations through an angle
8 = 90° which do this. We now consider the problem of finding
the geometrical locus of the centres of such rotations.
Problem 47. Given are two perpendicular lines g and g' which
intersect at the point S. Find the geometric locus of the centres
of all rotations which map g onto g' (see Figure 46).
62
TRANSFORMATION GEOMETRY
9*
Fig.46
Every rotation mapping g onto g' maps a line segment PQ
on ^ onto a line segment P'Q', of the same length, on g'. In
drawing P' and Q' we have to take into account the direction of
rotation. The rotation is completely determined by the choice of
such a pair of line segments. The centre of rotation, O, is the
intersection of the mediators of PP' and Q Q\ The fact that
9 = 90° means that O lies on the circles having PP' and QQ' as
diameters. These two circles, moreover, pass through S. Hence,
we have
LOSQ m lOQQ' = 45°.
The geometric locus of the centres of all the rotations through
an angle = 90° which map g onto g' is, therefore, one line
bisecting the angle between g and g'.
There are also rotations with 6 = — 90° which map g onto
g' ; the geometric locus of their centres is the other bisector.
Problem 48, This is a generalization of Problem 47. Given are
two lines g and g' which now meet at an arbitrary angle iff. Find
the geometric locus of the centres of all the rotations mapping
g onto g'.
Problem 49. Show that the set of all translations and half-
turns forms a subgroup SB' of SB.
The proof of this should present no particular difficulty.
ROTATIONS
63
We denote the fact that the elements of SB' are also elements
of SB by writing SB'c SB.
If we again denote the set of all translations by I, then we
have the following chain of inclusions:
Icgj'cSB.
We conclude our study of the group SB' by considering some
typical problems.
Problem 50. Show that the product of an odd number of half-
turns is always a half-turn.
Hint: use the fact that the product of two half-turns is a
translation.
Problem 51. Given five points P, Q, R, S, 7", construct a pentagon
having these points as the mid-points of its sides.
The vertices of the pentagon have been denoted by F, F lt F 2 ,
F 3 , F 4 (see Figure 47) Fis mapped onto itself by the following
chain of half-turns:
64
TRANSFORMATION GEOMETRY
ROTATIONS
65
II p Hq
H a
>F 3 >F 4 >F' » F.
//*
.Tit
Fis, therefore, a fixed point of the mapping
CI = rliH^ti^MnMp^
From Problem 50, £1 is a half-turn. Its centre, F, can be con-
structed by mapping any point A in the plane by CI. F is then
the mid-point of A A'.
The fixed point F can also be obtained without mapping a
figure (cf. Problem 15 and Figure 22).
We note that this problem always has a solution.
Problem 52. Construct an n-sided polygon, when the mid-
points Jl/ ( (i = 1, . . . , «) of all the sides are given.
This is a generalization of Problem 51. Let us denote the
half-turn about M t by H i and consider
£2 = H„H„^ , . , . H 2 H l .
n is a translation or half-turn according to whether n is even
or odd. This leads to two completely different solutions. If n is
odd, then the problem always has a unique solution just as in
the case n = 5.
For even «, however, the problem has a solution only if Q
is the identity; otherwise CI has no fixed point. But if £1 = /,
then every point in the plane is a fixed point; hence, in this
case there are infinitely many u-sided polygons with the given
points as mid-points of sides.
Consider in the two simplest cases, « = 4 and n = 6, what
condition is imposed on the given points by demanding that
Cl = I.
If we now apply Theorem 4 to the three left-hand factors we
obtain:
H D H E H C H C H B H A = / <*" H D H E H B Hji = I.
This last relation holds if, and only if, A, B, E, D arc the
vertices of a parallelogram (which may be collapsed onto a
line).
The position of the point C, therefore, is arbitrary and A, B,
E, D are the vertices of a parallelogram.
Problem 54. Given are two lines a, b and the three points P, Q
and R. Construct a path consisting of three line segments
which has the properties:
(a) the path starts on a and ends on b,
(b) P, Q and R are the mid-points of the three segments.
Problem 53. Five half-turns satisfy the relation
H e HqH c H b H a = H c .
What geometric condition on the disposition of the points is
characterized by this relation?
We first rearrange the relation:
Ff E H a HcH B H A = H c o H c H e H d H c H b H a = I.
THE GROUP OF ISOMETRIES
We now study the set ft of all isometrics, that is, of all transfor-
mations which preserve congruence, ft is the set of all finite
products of reflections:
Q = M t M t ...Mr M { .
£5 is a direct (orientation preserving) or an opposite (orientation
reversing) isometry according to whether n is even or odd.
Problem 55. Show that the set ft under the operation of com-
bination of transformations is a group.
ft is called the group of plane isometrics.
We have caUed figures congruent if they can be mapped onto
each other by mappings in the group ft. This notion of con-
gruence has the following remarkable properties.
1. Every figure is congruent to itself: F t s F t (property of
reflexivity).
2. Iff, is congruent to F 2t then F 2 is congruent to F t and vice-
versa :
F t St F 2 o F 2 = F, (property of symmetry).
3. If jF t is congruent to F 2 and F 2 is congruent to F 3 , then F, is
congruent to F 3 :
F, s F 2 and F 2 s F 3 s> F x at F 3 (property of transitivity).
Definition. The properties of reflexivity, symmetry and transiti-
vity characterize an equivalence relation.
66
THE GROUP OF ISOMETRIES
67
Equivalent figures form what is known as an equivalence
class.
As examples of equivalence classes with respect to con-
gruence, we have:
the set of all squares with sides of a given length d;
the set of all pairs of points a distance d apart;
the set of all right-angled triangles with sides 3 in., 4 in., 5 in.
Problem 56. Show that the concept of parallelism defined on
p. 28 is an equivalence relation.
We now want to give a complete description of the mappings
in ft. We first consider the direct isometries and show:
Theorem 10. A direct isometry with two distinct fixed points is
necessarily the identity transformation /.
P=p'
F,
P
Assume that 4> has the two fixed
points Fi and F 2 (see Figure 48).
Then the line/ joining F, to F 2 is — pf
obviously a poinlwise fixed line.
Now let P be an arbitrary point in
the plane and h be the perpendicular
from P to /. Since the right-angle
between / and h is left unchanged
by the mapping $, we conclude
that h is also a fixed line. There Fig. 49
are now two possibilities to con-
sider: either P' = P or P' and P are symmetrically placed with
respect to /. This latter possibility is ruled out since * is an
orientation preserving isometry. Hence, all points of the plane
are fixed points, that is, <J> = 1.
Now take an arbitrary direct isometry Q, that is, a product of
an even number of reflections. Let A'B' be the image under il of
a line segment AB. Problem 34 tells us that there are two re-
flections M p and M q such that M q M p maps AB onto A'B' (see
Figure 34). Hence, M p M q Q is a direct isometry with the two
68 TRANSFORMATION GEOMETRY
fixed points A and B and is, therefore, the identity. However,
MpMqQ = /<*>£1 = M q M r
We state this as:
Theorem 11. Every direct isometry can be expressed as the
product of two reflections and is therefore a translation or a
rotation. The direct isometries are the elements of SB. SB is
known as the group of direct isometries.
THE GROUP OF ISOMETRIES
69
V
1
A
*
A>
—
f
h
Fig. 49
Now we turn to the opposite isometries. We start with a
simple example, the mapping
* = M h M„M f ,
where/ and h are perpendiculars to g (see Figure 49).
Remembering that the order in which two reflections are
performed can be interchanged if their axes are orthogonal, we
deduce that
<fc = M h M g M t = M h M f M B = M t M h M P
But M h M f = 7" is a translation; hence we can write
<!> = TM g = M a T.
Our orientation reversing isometry is, therefore, the product
of a reflection and a translation with a vector v parallel to the
axis of reflection. We call this type of opposite isometry a
giide-refiection. For v = this reduces to a reflection. Reflec-
tions are, therefore, special cases of glide-reflections.
Theorem 12, Every opposite isometry is a glide-reflection.
2.
'<
Fig. 50
In order to prove this surprising fact we consider an opposite
isometry Q. Let A'W be the image of a line segment AB under
Q (see Figure 50).
According to Problem 34, we can find two lines p and q such
that M q M p also maps AB onto A'B'. We now introduce the
line r which joins A' and B'. We easily verify that M p M q M t Cl is
a direct isometry with the two fixed points A and B. From
Theorem 10, it must be the identity /:
M p M q M r Q = /•
Q - M r M q M p .
Hence, £2 can be written as a product of three reflections. It
remains to show that every product of three reflections is a
glide-reflection.
We have to distinguish between the two cases:
(a) p, q, r are parallel. In this situation it is easy to see that £1
can be reduced to a single reflection.
(b) p and r are not both parallel to q. Let us assume that q and
r intersect in a point R. Then M r M^ represents a rotation which,
according to Theorem 7, can also be written as a product of
70
TRANSFORMATION GEOMETRY
L
Fig. 51
two different reflections. Wc choose s perpendicular to p and a
line h such that M r M q = M h M t . We then have
Q = M r M*M p m M„M s M p = M h H s .
Now split the half-turn H s into two reflections M s and M 3 ,
where /is parallel to, and g perpendicular to, ft. This gives us a
representation
fi = M h H s = M„M t M p
which can immediately be seen to be a glide-reflection with the
axis g.
Now we can state:
Theorem 13. The group ft contains only four types of mappings,
namely translations, rotations, reflections and glide-reflections.
Note that Theorem 13 also tells us that every mapping Q of
ft can be represented as a product of at most three reflections.
Problem 57. What type of glide-reflection is CI = M r M q M f , if:
(a) p, q, r form an equilateral triangle ;
(b)p, g, r form a right-angled triangle;
(c) p, q, r all pass through the point S?
It is to be noted that a glide-reflection with a fixed point is a
reflection.
THE GROUP OF ISOMETRIES
71
Problem 58. Show that, if the lines p, g and r form a triangle,
then the axis of the glide-reflection £2 = M,M q M 9 passes
through two of the feet of the altitudes of the triangle.
Problem 59. Two congruent line segments AB and A'B' are
given. Construct the axis, g, and the vector, v, of the glide-
reflection which maps A onto A' and B onto B'.
One way of finding g and p is to express £1 as a product of
three reflections as in Figure 50, and to reduce this to another
triple product as in Figure 49. There is, however, a simpler way.
If we choose the line / in the representation H = M h M t M f
= H H Mf, given in Figure 49, to pass through A, then A is a
fixed point of M } . Hence A' is the image of A under H s . We
deduce that the mid-point of AA' must lie on the axis g (see
Figure 52). Using this property we can immediately construct
g from the two pairs of points A,A' and B, B'.
^v
\
\
—
\
\ A 1
Fig. 52
Problem 60. Investigate the following isometries:
(a) M t H Sl if S lies on g ;
(b) M g H St if S does not lie on g ;
(c) H A M f H A ;
<d) M„H F M .
Discuss the decomposition of half-turns into suitable products
of reflections.
Problem 61. Verify the following equivalence:
(M g H s ) 2 = /■**> S lies on g.
72 TRANSF0RMAT70N GEOMETRY
THE ROLE OF REFLECTIONS AND HALF-TURNS
IN THE GROUP St
The group St contains two types of involutions: reflections
and half-turns. These simple mappings bear a close relation to
the basic elements of plane geometry: lines and points. Corres-
ponding to every point and every line there is a unique trans-
formation. A geometric relation of incidence between points
and lines corresponds to a group relation between the corres-
ponding transformations. For example in Problem 61 we found
the equivalence
1 . M S H S M S H S = / *> S lies on g.
Another example of equivalent statements is
(F, C, //, A'are the vertices of a paral-
2. H K H H H G H F = lo jlelogram (which may be collapsed
lonto a line). (See Figure 22.)
We now wish to look for other equivalences of this kind.
(M t M q M p ) 2 = I means that the opposite isometry M^M^M,,
is a reflection M f ; hence M^M F m M r M f . The mapping which
is here described in two ways is either a rotation or a translation.
In the first case p, q and r all pass through a point O, in the
second case the three lines are parallel. In either case p, q, r lie
in a pencil of lines.
We have, therefore, the equivalence
3. {M r M q M^ 2 = Iop,q, r belong to a pencil.
If F is the mid-point of the line segment AB, then H F H A and
H B H F both represent the same translation. Hence
4. H F H B H r H A m lo Fis the mid-point of AB.
Problem 62. Verify the following equivalences:
5. M f H B MjH A = /-*>/is the mediator of AB;
THE GROUP OF ISOMETRIES
73
6. M f M b M f M a = / of is a bisector (or mid-parallel) of a
and b;
7. (H F H B T(H F H A r = I\ fFdivid.
(m,n positive integers)] jternally
divides the segment AB in-
in the ratio n:m;
8. (H B H F )"(H F H A ) m = l\ <s> fF divides the segment AB ex-
(/n,« positive integers)] <: * jternally in the ratio mm;
9. H B M f H B H A M s H A = lofis parallel to the line AB;
10. (H B MfH A ) 2 — lof is perpendicular to the line AB;
11. H S H C H S H B H S H A = /■**• S is the centroid of the triangle
ABC.
All the group relations on the left-band side have the same
structure; they are products of transformations representing
the identity. We shall call such products cycles of reflections and
half-turns.
These equivalences enable us to translate geometric relations
concerning position into algebraic language, and to translate
back. This idea leads to a new method of proof for geometric
theorems. The following problem serves to introduce this
method.
Problem 63. Show that a line
through the mid-points of two
sides of a triangle is parallel to
the third side.
We begin the proof by trans-
lating the given geometric re-
lations of position into algebraic
relations:
Fig- 53
Fis the mid-point of AB o H F H B H F H A = /;
G is the mid-point of AC o H G H C H G H A =* /.
74 TRANSFORMATION GEOMETRY
From these we deduce
— HqHfHgHaHa,Hf
= (H G H F ) .
(using Theorem 4)
The result H C H B = (H G H F ) 2 means that BC ■ 2FG. We
have proved even more than was required, for we have also
shown that the segment FG is half as long as BC*
Here we have proved a theorem in geometry by means of
algebraic manipulations with half-turns. This method can be
extended and other groups of mappings can be used to provide
proofs of geometric theorems.
Problem 64. Prove, using reflections and half-turns, that the
mediators of the sides of a triangle arc concurrent.
We restrict ourselves here to giving an outline of the solution.
Let /and g be the mediators of the sides AB and AC, and let a
be the line joining D, the point of intersection off and g, to the
vertex A. Then M g M a M f is obviously a reflection in a line. It is
then easy to show that the axis of this reflection is the third
mediator.
We conclude our study of the group of isometries with the
following remarks. It is now possible, using the knowledge of
mappings and groups which we have gained so far, to define
the somewhat vague idea of geometry more exactly. Geometry,
in the sense of Felix Klein, is a study of those quantities and
properties of figures which are left invariant under all the
mappings of a group; it is the study of invariants of a certain
group of mappings. For every group of mappings we can,
therefore, develop a particular geometry.
Corresponding to the group ft of isometries is the so-called
geometry of congruences, which formed the main part of the
traditional elementary geometry course.
* It should be noted that this is a special case of the theorem that rays
through a given point are divided by parallel lines in equal ratios. No
assumptions about continuity arc required for our proof of this special
case.
THE GROUP OF TRANSFORMATIONS
MAPPING A SQUARE ONTO ITSELF
The group ft contains translations, rotations, reflections and
glide-reflections.
We now wish to enumerate all the iso-
metries which map a square ABCD onto
itself. Obviously, no translation or glide-
reflection can map the square onto itself,
and so wc need only consider rotations and
reflections. It is easy to see that the square
is mapped onto itself by the eight isometries
A=A
a=e' shown in Figure 54.
Identity I
-9 9
C-A' D=A>
-0 Q-
A=0' BzA' AzC' B=D'A=B ! S=C'
Rotation Rotation Rotation
through* SO' t hrough +100' through* 270 '
ff, ff z R 3
D=D' C=A' D=B' C*C's D=A' C=B' D*C> , C=0'
A=C>
b=b' a=a' b=d' a=d' b=c' a=b' b*a'
Ml
M,
M 3
Reflections about the axes shown
75
M t
Fig. 54
76
TRANSFORMATION GEOMETRY
There are no other isometries which leave the square invariant
and so the product of any two of these mappings must again be
one of the eight. For the same reason, the inverse of each of
these mappings must also belong to this set of eight. Therefore,
we have a group which has only a finite number of elements.
We have already met finite groups defined by means of numbers
and so it should not surprise us to find that finite groups also
occur in transformation geometry.
Before discussing this group further we wish to know how we
can find the product of any two of its elements. This will con-
siderably simplify the working out of the group table. The map-
ping R 2 > f° r example, maps A onto C, B onto D, C onto A and
D onto B. We express this by writing
similarly,
(A B C D\
* 2 = \c DA b\-
(A B C D\
3 ~ \d c b a y
B C
C B A
For the mapping M 3 R 2 (first R 2t then M 3 ) we have
A B C D
R 2 I i i i
C D A B
M 3 i i, I I
B A D C
that is,
M
* " (j
I C £>j
[b a d cy
This, however, is the mapping M A . Hence, we get the relation
M 3 R 2 = M A .
If, in this way, we work out all the products we get the
following table.*
* Note that wc find the product M 3 R 3 by looking in the row labelled M 3
and the column headed /?..
THE
GROUP
OF
TRANSFORMATIONS
/ R t
Ri
*3
M x M 2 Mi Af 4
/
I Rr
Ri
*3
| M t M 2 Mi M A
*l
Ri Ri
Ri
/
1 M 3 M A M 2 M x
R 2
R 2 R 3
I
Ri
] M 2 M t M A M 2
R 3
Ri I
*i
R2
1 A/« M 3 Mi M 2
M x
M % M 4
M 2
Mi
'/ R 2 Ri Rt
M 2
M 2 M 2
M v
M 4
\R 2 I Ri Ri
Mi
Mi M t
M A
M,
,Ri R 3 I R 2
M A
M A M z
Mi
M t
| ^3 -Ki R2 *
77
This group table has the following remarkable properties.
In every row and every column, every element (mapping)
occurs exactly once. This is a property of all group tables.
I and the elements R t , R 2 and R 3 form a group by them-
selves (see the top left-hand quarter square of the group table).
That is, our group of eight elements contains a subgroup of
four elements consisting of those mappings which preserve the
orientation of the square. Here again we learn a general fact
about groups of mappings : if we demand additional invariants,
then the group is restricted to a subgroup.
The group is non-commutative. It does contain pairs of
elements which commute, but we have, for example,
M 2 R 3 = M A and R 3 M 2 = M 3 .
Since group multiplication is associative, we can work out
higher products. This corresponds geometrically to the product
of more than two mappings. For example.
MiR 3 M 2 R 3 M A = (M \R 3 )M 2 (R 3 M A )
= M 3 (M 2 M 2 )
= M 3 I
= M 3 .
Every other way of working out this product leads to this
same result. This is the content of the associative law, that is,
the group postulate IV.
Hence, we get the same result by the following bracketings of
the factors:
78
TRANSFORMATION GEOMETRY
THE GROUP OF TRANSFORMATIONS
79
M 1 R 3 M 2 R 3 M 4 = (M 1 R 3 )(M 2 R 3 )M 4 = (M 3 M 4 )Af 4 = R 2 M A
= M 3 ;
M 1 R i M i R s M 4 , m A/,i? 3 Af 2 (/? 3 A/ 4 ) - M l R 3 {M 2 M 2 )
= M t R 2 = M 3 .
We stress once more that the operation of combining trans-
formations is always associative.
We shall now demonstrate a general theorem of group theory
by considering our finite group. Denote our group of isometries
by © 8 . We can see immediately that the two elements / and R 2
form a subgroup § 2 . If we now multiply both elements of & 2 by
an element of <5 8 not belonging to § 2 — say by M t — we obtain
the set of elements
M& 2 = {M t I, A/,/?,} = {M u M 2 }.
These are two elements of (5 8 not belonging to § 2 « If we take
another clement of <§ 8 which has not occurred so far, for
example, M 3 , we obtain in the same way
M& 2 = {M 2 l, M 3 R 2 ) - {M 2 , M 4 }.
If we repeat this process with one of the remaining elements
of <5 8 , for example, R lt we obtain
*i$2-{*iA*i*a} = {*i,* 3 }.
Now we have a complete partitioning of the elements of (<5 8
based on the subgroup |> 2 . We may speak of it as a decomposi-
tion of (5 8 with respect to § 2 , and we write
62 = &* + {Mi, M 2 } + {M 3I Mt} + {R U R 9 ).
The elements of <5 8 by which we 'multiplied 1 § 2 are not
uniquely determined. We remark, however, without proof that
the decomposition of © 8 with respect to §» 2 is, nevertheless,
unique ; we always obtain the same four classes. The essential
fact is that <5 8 can be divided into a number of classes which all
contain the same number of elements, namely the same number
of elements as are contained in § 2 . This is true for every sub-
group § of a finite group (5. After a finite number of steps the
process of splitting into classes terminates and we always
obtain a number of classes with the same number of elements in
each of them. So we have verified in a special case the general
statement:
If a finite group © has a subgroup §, then the number of
elements in § is always a divisor of the number of elements in
©.
Our group © 8 can, therefore, only contain non-trivial sub-
groups with 2 or 4 elements. The following is a complete
enumeration of the subgroups of © 8 .
{/, R 2 }, {I, A/,}, {/, M 2 }, {I, M 2 }, {I, M 4 },
{/, R u R 2 , R 2 }, {I, R 2 , M lt M 2 ], {I, R 2 , M s , M A ).
The subgroup {/, M x } is characterized geometrically by the
fact that the vertices B and D are fixed points under mappings
of this group.* Try to characterize the remaining subgroups of
<5 S in this way.
According to the particular field of mathematics which has
given rise to the group in question, this theorem contains a
number theoretic, algebraic or geometric statement— the group
concept reveals common mathematical structures.
We now wish to connect our group © 8 with a group of
permutations. Here we touch upon the field of mathematics
where the idea of a group first appeared.
If we only consider the four vertices A, B, C, D of our square,
every isometry in © 8 produces a one-one mapping of the point
set {A, B, C, D} onto itself. We have already mentioned on
p. 19 that such one-one mappings of a finite point set onto
itself are called permutations.!
Our group of isometries which leave the square invariant
consist of permutations which preserve relations of proximity.
That is, in the square the vertices A and C are adjoining B; this
property is preserved by mappings of © 8 . There are other
• This is another example of the previously mentioned fact that the choice
of a subgroup corresponds to an increase in the invariants. Now not only
is the square as a whole invariant but also two vertices slay fixed.
t Often the actual result of the mapping, that is, a particular order of the
elements, is called a permutation.
80 TRANSFORMATION GEOMETRY
permutations of {A, B, C, D} apart from the eight in <5 8 . For
example, the permutation
v-i A B C D \
\b da cj
does not occur in (5 8 ; after this mapping C and D are the ad-
joining points to B (we must think of the cyclic order of the
four points). Altogether there are as many permutations of four
elements as the number of different orders of these elements.
This number is
4.3.2.1 =4! = 24,
for we have four possible choices for the first place, for the
second place there are three choices, for the third place we have
two elements to choose from, and the remaining element goes
into the fourth place. The 24 permutations of the four elements
A, B, C, D can be characterized by the second line of our
permutation symbol; in alphabetical order they are:
ABCD (r)
BACD
CABD
DABC (r)
ABDC
BADC(m)
CADB
DACB
ACBD
BCAD
CBAD (w)
DBAC
ACDB
BCDA (r)
CBDA
DBCA
ADBC
BDAC
CD AS (r)
DCAB
ADCB(m)
BDCA
CDBA
DCBA (m)
These permutations form a group of which our group of
mappings, © s , is a subgroup. We verify again in this example
that the number of elements in a subgroup is a divisor of the
number of elements in the full group; 8 is a divisor of 24. In
the table of the 24 permutations, the rotations are denoted by
(r) and the reflections by (m).
Problem 65. Find the group table of the isometries which map
an equilateral triangle onto itself.
Problem 66. The two permutations
(ABCDEFGH)
1 ~{C H D F B A G E )
THE GROUP OF TRANSFORMATIONS 81
(ABCDEFGH)
and ¥, = < y
{CDABEFGH)
are elements of the group of permutations of eight objects.
(a) Determine the permutations *Pi -1 , x ¥ 2 ~ l , *F 2 2 .
(b) Determine the permutations ^^^ ^j^ and V^ 2 '¥ t ~ 1 .
(c) Show that H? 2 is an involution;
(d) Which is the smallest exponent p for which *iEV = /?
(e) Show that the elements 4\, «F, 2 , ^j 3 , . . . , T^ -1 , ¥^ = /
form a group.
For the solution of
(d) consider a set of
eight points and indi-
cate the mapping de-
fined by Tj (see Figure
55). From the cycles
appearing in this figure
we can easily compute p. Fig.55
Problem 67. Consider each capital letter of the alphabet in
turn and find the group of isometries which map the figure onto
itself. Divide the letters into classes by putting all letters having
the same group of isometries into the same class.
HHHHHHHH
bpbpbpbp
FFFFFFFF
EEEEEEEE
TTTTTTTI
riTlTITl
NNNNNNNN
ZTZTZTZT
Fig 56
VTVTVTVT
LL LLLLLL
BO 69 BS 83
HHHMHHHM
Problem 68. Repeat Problem 67 for the frieze patterns shown in
Figure 56 which we imagine to be extended indefinitely in both
directions.
Problem 69. Verify the relation
(OT)" 1 =*P- 1 *'" 1 ,
which holds for all groups, and give some examples taken
from the groups we have considered.
F
ENLARGEMENTS
Fig. 57
Let 5 be a fixed point in the plane and ft a non-zero number
(positive or negative). We define a mapping of the plane by the
following rules:
A point A and its image A' lie on the same straight line through
S, that is, A, A' and S are collinear.
Moreover, each pair A, A' satisfy the relation
SA'
-=-r = p (here we think of directed line segments).
This mapping is called an enlargement, S is the centre and /i .
the scale factor of the enlargement. If n>0, the corresponding
points A and A' lie on the same side of S. If, however, ^<0,
then S lies between A and A'. An enlargement with scale factor
- 1 is the same as a half-turn about S.
If we are given S, A and A', then it is easy to construct the
image of any other point B (see Figure 57). We draw the line
through A' parallel to AB to intersect SB in B'.
We then have,
SB' SA'
SB ~ SA ~ **'
82
ENLARGEMENTS
83
It is just as easy to find the original point B if we are given the
image point B". The construction shows that enlargement is a
one-one mapping. In general, S is the only fixed point of the
transformation.
Here is a list of other properties of enlargements which will be
used in later constructions.
From Figure 57 we can see that if a point P moves on the
lineg, only the ray of projection SP' will change in the drawing.
In particular, the parallel g' remains fixed. This means that if P
moves along g, its image P' will describe the line g';g' is, there-
fore, the image of g.
1. Enlargements are line-preserving transformations. More-
over, a line g and its image g' are always parallel.
We note that lines through S are mapped onto themselves,
although not all their points stay in the same place. They are
fixed lines but not pointwise fixed.
As each pair g, g' and h, K of corresponding lines is parallel,
the angle between g and h is preserved by the transformation ;
in other words:
2. Enlargements are angle-preserving transformations.
We have, further, the relation
A'B'
S A1
SA
AB
= /*>
which means:
3. The lengths of corresponding line segments are in the ratio
l:/i.
If fi>0, corresponding line segments are parallel in the same
sense; if #<0 they are parallel but are in the opposite sense.
We deduce immediately the following property:
4. The ratio of the lengths of two image line segments is equal
to the ratio of the lengths of the original line segments.
Thus, enlargements preserve ratios of distances.
84
TRANSFORMATION GEOMETRY
In a triangle and its image under enlargement the altitudes
are also corresponding line segments, hence the areas are in the
ratio 1 :/?. Since polygons can be divided into triangles, and
figures with a curved boundary can be approximated arbit-
rarily closely by means of polygons, we have:
5. The areas of a figure and its image under an enlargement are
in the ratio 1 ip 2 .
Further, we have:
6. The inverse transformation to an enlargement E(S,n) is the
enlargement £" '(5, l//i).
We shall now look at a series of problems which can be solved
by enlargements applied to a geometric locus.
Fig. 58
Problem 70. Construct a line through one point of intersection,
S, of two circles T, and F 2 , such that the two chords on this
line have lengths in the ratio 2:3.
Let X and X' be the other points of intersection of the
required line with the circles I\ and F 2 (see Figure 58). Then
we can write
SX' _3
SX 2'
ENLARGEMENTS
85
Hence, X' is the point of intersection of T 2 with the image of F t
under the enlargement with centre 5 and scale factor — 3/2.
The enlargement with scale factor 3/2 gives another solution ;
in this case one of the chords lies on the other.
Problem 71. Given are two squares with a common vertex A.
Draw a line through A such that the lengths of the segments
inside the squares are in the ratio 1 :2.
Problem 72. Given are two circles T^ and r 2 , and a point S.
Construct a point A on Fl and a point B on T 2 such that S is the
mid-point of AB.
Problem 73. Construct the triangle ABC, given the lengths of the
sides AB and AC, and the length of the bisector of LB AC.
Let AB = c, AC = b and let the length of the angle bisector
A W be w. We begin the construction by drawing A W. A geo-
metric locus for B is the circle F with centre A and radius c
(see Figure 59). Now we know that W divides BC in the ratio
c:b. An enlargement with centre W and scale factor -b[c will,
therefore, map F onto a geometric locus for the vertex C. On
the other hand, C must lie on the circle with centre A and radius
b.
86
TRANSFORMATION GEOMETRY
Problem 74, Construct a triangle ABC in which BC is of length
a, the median from B is of length m, and the lengths of the sides
AC and BA are in the ratio 1 :X.
We start by drawing the line segment BC. The Apollonius
circle T defined by BP:PC = X:l is then a geometric locus for
A. An enlargement with centre C and scale factor i maps T~
into a geometric locus for the mid-point A' of AC. However,
A' also lies on the circle with centre B and radius m.
Problem 75. Construct a triangle ABC given the lengths of the
sides AC and AB, and the length s of the median from A.
We begin the construction by drawing AB = c. The circle T
with centre A and radius b {— AC) is then a geometric locus for
C. This can be transformed into a locus for C\ the mid-point
of BC, by an enlargement with centre B and scale factor ■£.
Another locus for C is given by s.
If we begin the construction by drawing the line segment
AC, then the problem can be solved by using a half-turn.
In the following examples enlargements are used in an
essentially different way. The common method of solution is by
relaxing one, or possibly more, of the conditions. This is done in
such a way that the remaining conditions determine a family
of figures which can be mapped onto each other by enlarge-
ments. After constructing one member of this family, we trans-
form it by an enlargement so that its image satisfies the omitted
conditions as well. Thus, the scale factor of the enlargement
used depends on the choice of the representative, that is, upon
the way we draw the initial figure.
ENLARGEMENTS
87
Problem 76. Given a triangle ABC, construct a square with two
vertices on BC and one vertex on each of the sides AB and AC.
If we ignore the condition that J should lie on AB, a family of
squares is determined. The square R'S'T'U' (see Figure 60) is
one representative of this family. By a suitable enlargement
with centre C, we can map 7" onto T without disturbing the
other incidences. The scale factor of this enlargement depends
upon the choice of R'S'T'U'.
We leave it to the reader to show that the problem could also
be solved using enlargements with centre A and with centre B.
Problem 77. Draw a circle to pass through two given points A
and B and to intersect the given line s in a chord which sub-
tends an angle of 120° at the centre of the circle.
The centre M of the circle to be constructed lies on m, the
mediator of AB (see Figure 61). We first consider the family of
all circles with centres on m which satisfy the condition on their
intersection with s. This is a family of circles which can be
mapped onto one another by enlargements with centre S, the
point of intersection of m and s. By a suitable mapping of a
representative circle T' we can obtain a circle T which satisfies
all the given conditions. Note that the problem has two solu-
tions, since there are two points A t ' and A' 2 on T' which
can be mapped onto A. Figure 61 shows only one solution.
Fig 6i
THE GROUP PROPERTIES OF ENLARGEMENTS
We now investigate the transformation obtained by com-
bining two enlargements. Assume that:
88 TRANSFORMATION GEOMETRY
E t is an enlargement with centre S t and scale factor ft Y ;
E 2 is an enlargement with centre S 2 and scale factor n 2 .
s=s,=s'
Fig- 62
A point A is mapped onto A t by E lt and A t is mapped onto
A' by E z (see Figure 62), The transformation E 2 E l leaves the
line s through S x and 5 2 fixed, since s is a fixed line for both
enlargements. The image of an arbitrary line h through A is
parallel to h and through A'. From this it follows immediately
that the line g, joining A to A', is another fixed line of E 2 E lt
The point S where the lines g and s meet is, therefore, a fixed
point of E 2 E X . We now want to calculate the position of S on s.
In order to do this we introduce the following vectors:
SiS 2 = rf, S 2 S' = p (S' is the image of S under E{),
SiS = x t S 2 S = q.
Then we have the following relations :
p+d = HiX, q = fi 2 p, d+q = x.
From these we obtain
x = d+q = d+ft 2 p = rf-t-AftiLX-O = (l-^d+tt^x.
Solving this vector equation with respect to x, we obtain
x - £=&_ d *
I-H,fi 2
* The position of S relative to S, and 5, can also be represented by the
fraction X = f£. We find A = *~^ .
ENLARGEMENTS
89
This shows that 5 does not depend upon the choice of A ; S is
determined by the two mappings E l and £ 2 . For the transforma-
tion E 2 E X , therefore, corresponding pairs of points A, A' lie
on lines through the fixed point S.
From Figure 62 we can also recognize the following relation :
SA' _ S'A l SA'_
SA ~ SA 'S'At V^ 2 '
Hence, we have shown that E 2 E t is an enlargement with
centre S and scale factor « = PiH 2 .
The product of two enlargements E x and E 2 is, in general,
again an enlargement, E. The three centres S lt S 2 and S are
collinear, and the scale factors satisfy the relation ft = fixfi 2 .
In Figure 63 this result is illustrated by the transformation of
the triangle ABC. A new group of mappings begins to be recog-
nizable.
Fig 63
Problem 78. Given are two enlargements E^S^) and
E 2 (S 2 , -2/3). Construct and calculate the position of S, the
centre of the mapping E 2 E l .
90
TRANSFORMATION GEOMETRY
To construct S, map an arbitrary point A (which does not
lie on the line SiS 2 ) by £ 2 £i- The position of S can be calculated
by the vector formula given above. Here we have
l-(-2/3) 5
I-2<-2/3) a 7
Problem 79. Find the product of the two enlargements £[(5, ,2)
and E 2 (S 2 ,i).
If we apply the transformation E 2 E X to a point A we obtain a
situation like that shown in Figure 64. It can be seen that for
every pair of corresponding points A and A', the vector from A
to A' is v = id. Hence, E 2 E t is obviously a translation with the
vector v.
Since, for our pair of enlarge-
ments, fi x fi 2 = 1, the denominator
in our formula for x vanishes. If
the value of /i^ is ver Y c l° se t0
1, then the vector x will be very
long. We can take fi l fi 2 = 1 to
characterize the point at infinity
on the line through S, and
s * f i 2 Fig. $4
We can now consider translations to be special cases of
enlargements ; the centre of enlargement for a translation is a
point at infinity and the scale factor is 1.
Problem 80, Two enlargements £ t and E 2 are given with
/ii/t 2 = 1. Show that v, the vector of the translation E 2 E U is
given by
1
» = (i-£>-a-i'j*
Problem 81. T is a translation with vector v and E an enlarge-
ment with centre S and scale factor /i#l, Show that ET and
TE are enlargements with scale factor ft and construct the
centres of these transformations.
• The point at infinity is not a proper point. We consider it as an ideal
element corresponding to the value X = 1 for the ratio of distances of a
point from two given points.
ENLARGEMENTS
91
We can use the same method as in the case of the two finite
enlargements. The line through S parallel to v is a fixed line of
TE = £' and ET = £". In Figure 65 the centres S' of £' and
S" of E" have been obtained by finding the images of a point
A. It is easy to verify that
x < = — I „ a nd x" = 2-rir.
fi — 1 ft — 1
If we use the equivalence
£' = 7£ <=>£'£"' =7",
and the corresponding one for E", then these relations can be
deduced from the result of Problem 80.
Theorem 14. The set of all enlargements forms a group, 91,
under the operation of combination of transformations.
To prove this we must show that the group postulates are all
satisfied. When we consider the product of two elements of 21
we have to consider the possibility of the transformations
being translations.
The unit element of 21 is the identity /. As in the case of
rotations, / can be represented in different ways; every enlarge-
ment with ft m 1 represents / provided the centre of enlarge-
ment is a finite point.
The group is not commutative; this means that, in general, a
product depends on the order of the factors. We give as an
example of two elements which do not commute, the two
transformations J and £ of Problem 81. There are, of course,
92 TRANSFORMATION GEOMETRY
commutative pairs in 21 ; for instance, two enlargements with
the same centre always commute.
Problem 82, Show that a pair of enlargements, neither of which
is a translation, commute if, and only if, both mappings have
the same centre.
Two enlargements commute if E 2 E X and E X E 2 have the same
centre. This is obviously true if S x m S 2 , that is, if d — 0.
Now assume that d ^ 0. The centres of E 2 E X and E X E Z can
then be described by the two vectors
x > = l ~J}K d and x » = -±Zl±-d.
l-^iJ'2 l-JHth
x' and x" describe the same point if x'~ x" = d (see Figure
66), that is, if
1-^1^2 l-A»iA*2
This equation can be transformed to
(l-^)(l-,t 2 ) = Q.
But this can never hold since n Y # I and fi 2 ^ I.
If p x n 2 = If it can easily be deduced from the formula in
Problem 80 that the two translations E 2 E X and E X E 2 have
different vectors (unless lii+fi 2 = 2, in which case {i L = (i 2
= 1).
Fig, 66
ENLARGEMENTS
93
Problem 83. An enlargement E x with centre S t maps A onto A v .
A second enlargement E 2 with centre S 2 maps A x onto A^'.
Hence, A x is the image of A under the transformation E 2 E l .
Construct the image of A under the transformation E X E 2 .
We give the solution without comment in Figure 67.
Problem 84. If £, and E 2 are two enlargements, show that
EyE^E^y 1 = E^E.-'E^ 1
is always a translation T with a vector v which has the direction
of the line joining S x and S 2 .
This is illustrated in Figure 67; Tmaps the point A x onto
a 2 :
Try to express the translation vector of T in terms of fi u [t 2
and d = S X S 2 .
Problem 85, Two enlargements E X (S X , /*,) and E 2 {S 2 , ;< 2 ) are
given. Construct the centre of the mapping E t E 2 E l '.
Problem 86, Construct the centre of the transformation EiE 2 E u
where E lt E 2 and E 3 are three given enlargements, and illustrate
geometrically by means of this figure that the associative law
holds in St, that is
E 2 (E 2 E t ) = (E 3 E 2 )E X .
What incidences can be derived from this?
Problem 87. Prove the following theorem of Mcnelaus: If a
line s intersects the sides of a triangle A, A', A* in the points
Sj, S 2 , S 2 , then
o [ si & 2 A ^3^ ,
'S i ~A"S 2 ~A r "S^A* ~
To prove this we introduce the following three enlargements :
*(** = S$H** = Wr) E ( s ^ = !£}
The enlargements with centres on a line s form a subgroup of
9J; this is a simple consequence of the rules for composition.
94
We shall denote this subgroup by 9t s .
Then E 3 E 2 E 1 c 9tj. But A, which does not lie on s, is a fixed
point of this mapping, This can only happen if the mapping is
the identity. However,
£ 3 £ 2 £, = I ^Hrfhl** - U
which proves our theorem.
Problem 88. Prove the following theorem of Desargues: If the
lines joining corresponding vertices of the triangles ABC and
A'B'C are parallel, then the points of intersection of cor-
responding sides lie on a line s.
The proof, like that of Problem 87, can be reduced to the
composition of transformations in 91.
ENLARGEMENTS
95
Given two circles T t and T 2 there are, in general, two enlarge-
ments Ei and E 2 which map oue onto the other. The centres
Si and S 2 lie on the line through the centres of the circles;*
the easiest way of finding them is by using parallel radii of the
circles (see Figure 69). The corresponding scale factors are
**2 j r i
}ii=*~r and /*2 = _ r-
Problem 89. Show that the six centres of enlargement associated
with the three circles T u T lt r 3 , lie on four lines, three on each
line. (Theorem of Monge.)
We pick out two circles r f and F ; . There are two enlargements
£,j and E lt ' which map T, onto T } ; let E l} be the one with a
positive scale factor. We immediately deduce the following
relations:
£23^2 = £i3> £23^2 = £13 >
£23 £l2 = ^13 > ^23 £l2 = £l3>
which imply the statement to be proved.
Problem 90. There are two enlargements £ and £* which map
the circle T onto the given circle T', Describe the mapping
£*£ _1 .
The following four problems provide a method, based on
transformation geometry, for proving Feuerbach's Theorem on
the so-called nine-point circle.
Problem 91. Show that, in every triangle, the point of intersec-
tion of the altitudes H (the orthocentre), the centroid G, and
the circumcentre O lie on a line (the Euler line).
Hint: use the enlargement which maps the given triangle
ABC onto the triangle A'B'C formed by the mid-points of the
sides of ABC.
Problem 92. Show that the circumcentre O' of the triangle
A'B'C lies on the Euler line of the triangle ABC.
Problem 93. Denote the circumcircles of ABC and A'B'C by
r and T' respectively. They can be mapped onto each other
by two enlargements:
* S, and Sj are often referred to as centres of similitude.
96 TRANSFORMATION GEOMETRY
E(G,-i) and E*(G*&.
Show that the centre of enlargement C* is H.
Problem 94. The image of the triangle ABC under E* is a
triangle A*B*C*. A*, B* and C* are the mid-points of the
lines joining H to the vertices A, B and C.
Note that (A', A*), (B', 5*) and (C, C*) are pairs of corres-
ponding points under the transformation E*E~ l and use this
to prove that the circle F passes through the feet of the altitudes
of the triangle ABC.
The circle I"' has now been shown to contain the mid-points
of the sides, the feet of the altitudes, and the mid-points of the
lines joining the orthocentre to the three vertices— r' is Feuer-
bach's nine-point circle.
SUBGROUPS OF 9t
We have already mentioned that:
The enlargements with centres on a fixed line g form a sub-
group 9t, of 91, The transformations of 9l fl are distinguished by
the fact that they all leave the line g invariant.
If two enlargements have the same centre G, that is, if d = 0,
then x = 0. Hence, the centre of the product is G also. We
conclude that the transformations of 91 with a given centre G
also form a subgroup 9f c . Its elements are characterized by the
in variance of the point G. If C lies on g, then
that is, 9I C is a subgroup of 9t r
We noted earlier that subgroups can be obtained by demand-
ing additional invariants. 9t s is obtained from 91 by demanding
that the arbitrary lineg is invariant, and 9t G by demanding that
the point G is invariant.
We can also demand the invariance of a whole figure. We
may, for example, imagine the square and circle pattern shown
in Figure 70 to be continued indefinitely in both directions. This
leads to a figure which is invariant under enlargements with
centre S and scale factor (^/2)" where n is any integer. These
mappings form a subgroup 9t s ' of % s and therefore of 9t itself.
Unlike the previous examples 91/ is a discrete subgroup of 91.
ENLARGEMENTS 97
By this we mean that the transformations in 9t s ' do not form
a continuous family.
The most general enlargements leaving Figure 70 invariant
have scale factors ±(-j2)' 1 where n is any integer. These map-
pings form a group 9( s " and we have the chain of inclusions:
9l s 'c9I s "c9r s cfc.
All these subgroups of 91 have infinitely many elements.
SIMILARITIES
99
SIMILARITIES
6-45"
Fig, 71
Problem 95. A point A and lines b and c are given. Construct a
square with A as a vertex such that the vertex B lies on b and
the vertex Conc.C is the vertex opposite A.*
To solve this we first disregard the condition that C should
lie on c. We then get a family of squares of which AXX'D* is a
representative (see Figure 71). In this we can think of X' as the
image of Sunder a rotation about A through 45°, followed by an
enlargement with centre A and scale factor ,/2. If X' moves on
the line b', the image of b under these two transformations, then
C is the intersection of b' and c. The rotation with = —45"
leads to a second solution of the problem.
This method of solution is based on transforming a geo-
metric locus ; we have used this method before. What is new is
that here two different types of transformations have been
• Cf. Problem 23 in which similar conditions were given.
98
combined — the transformation used was the product of a
rotation and an enlargement having the same centre. It is easy
to see that the two mappings commute; it does not matter
whether the rotation or the enlargement is performed first.
We call the product of a rotation and an enlargement with
the same centre S a spiral similarity.
This defines a new type of transformation. A spiral similarity
is characterized by the centre S, the angle of rotation 6, and the
scale factor // ; we may denote it by iV(S,6,ft). Before we turn
to the theory of spiral similarities we give two further construc-
tion problems of the same type as Problem 95.
Problem 96. Given are a point A and two lines b and c. Con-
struct a triangle ABC with B on b, C on c, £_BAC = 60° and
LABC = 45°.
A procedure analogous to that used in Problem 95 leads to a
spiral similarity W(A, +60°, sin 45°/sin 75°).
Problem 97. Given are a point A, a circle T and a line g. Con-
struct a rectangle with sides in the ratio 1 :2, which has one
vertex at A, the vertex B on T and the vertex C (opposite to A)
on g. AB is to be one of the shorter sides of the rectangle.
The solution to this problem uses a spiral similarity
W(A,± tan" 1 2, JS).
The angle of rotation and the scale factor can be constructed in
a model figure.
Problem 98. Given are four points A, B, C, D. Find a rectangle
with sides in the ratio 1 :2, such that each side (or its extension)
passes through one of the given points.
This is a generalization of Problem 45. Now the strip bounded
by a and c is mapped onto the strip bounded by b and d (see
Figure 72) by the spiral similarity
W(M, 90°, 2).
The vector u is mapped onto a vector w which is orthogonal
to u and twice as long. The translation T with vector t maps the
image C (= W(C)) onto D. If we draw w with end-point D,
100
TRANSFORMATION GEOMETRY
then the starting point A* is a point of the side b of our rect-
angle. A* and B together determine b and it is then easy to
complete the rectangle.
In solving Problem 98 we had to use the product of a spiral
similarity and a translation. This means we multiplied an
enlargement by a rotation and a translation.
Definition. A finite product of isometries and enlargements is
called a similarity.
Similarities, like isometries, can be divided into two classes,
direct (orientation preserving) similarities and opposite (orienta-
tion reversing) similarities.
DIRECT SIMILARITIES
The spiral similarity is an example of a direct similarity. If
SIMILARITIES 101
R(S,0) is a rotation and E(S,p) is an enlargement, then
ER= RE = W(S,8,p)
is a spiral similarity with centre S, angle of rotation 0, and scale
factor fi.
Fig 73
For spiral similarities we have the relation
W(S,6,p) = W(S,0 +« -fi).
In order to obtain a unique representation, we shall always
assume that fi>0 in what follows.
Fig 74
Problem 99. Let K be a direct isometry and E an enlargement
with centre S and scale factor p. # 1. Show that the direct
similarity SI = EK always has a fixed point F.
102
TRANSFORMATION GEOMETRY
We first assume that K is a rotation with centre O through
angle (see Figure 74), We can always find a point F which is
mapped onto the same image F' by both K and £"'. To do this
we take two corresponding points A and A' (= E'^A)) and
draw the isosceles triangle with angle AO*A' = 0. The triangle
AO*A' is mapped onto FOF' by a spiral similarity with centre
S. F is now a fixed point of the transformation EK.
If K is a translation, then a fixed point can be constructed in
an equally simple way.
The definition of similarities may lead us to expect to find
many different types of transformations in the class of direct
similarities. Surprisingly, this is not so; we can show:
Theorem 15. Every direct similarity is a spiral similarity.
The proof depends on the fact that a direct similarity is
completely determined by two points A and B, and their images
A' and B'.
If the line segment A'B' is congruent to AB, then Q, the
transformation mapping AB onto A'B', is a translation or a
rotation. Both are special cases of spiral similarities.
We now assume that AB and A'B' are not congruent. Let/> be
the mediator of AA'. The reflection M p maps A onto A'; we
denote the image of B under this mapping by B*. We now
introduce g, the bisector of the angle between A'B* and A'B'.
The reflection M q leaves A' fixed and maps B* onto fit. If we
denote by E the enlargement with centre A' and scale factor
H = B'A'jB-\A', we obviously have
fi - EM q M p = EK.
AT is a translation or rotation according to whether p and q are
parallel or not (see Figure 75).
We now have the situation of Problem 99. We know, there-
fore, that the mapping Q. has a fixed point F. The isometry
K m M q M p can also be obtained as the product of two different
reflections; if we choose />, to be the line through O and F we
have to use as g { the perpendicular through to the line s
SIMILARITIES
103
joining A' and F,f In order not to overcomplicate Figure 75,
these extensions are shown in Figure 76. We now have
Q = EM^M, = EM,M Pi = E(M q MXM s M Pt ).
The introduction of M s 2 = / does not, of course, change the
transformation. M s M Pi is a rotation with centre F and M q M s
represents a half-turn, H L , which wc can also interpret as an
enlargement. Front the rule for combining enlargements, we
know that EH L is an enlargement with its centre on s. But F
and fare corresponding points for both E and H L ; hence EH L
leaves the point F fixed. Therefore, F is the centre of enlarge-
ment of EH L = E*.
Thus we have obtained a representation
a = E*(F,-pi)R(F,4>) = WW,-fi) = WPAft
Q is a spiral similarity with centre F.
Problem 100, A, A' and B, B' are corresponding pairs of points
for the spiral similarity
Determine the centre S, the angle of rotation 0, and the scale
factor fi.
t In the case of a translation, p v is the perpendicular to s through F.
104
TRANSFORMATION GEOMETRY
Fig, 77
8 appears as the angle between corresponding lines g and g',
and n = A'B'jAB.
If A and A' are corresponding points, then /_ASA' - 0.
Using this property we can find S by the construction indicated
in Figure 77.
The product of two direct similarities is again a direct simi-
larity. If we consider translations and enlargements as special
cases of spiral similarities, then we can state our result as fol-
lows:
Theorem 16. The spiral similarities form a group S under the
operation of combination of transformations.
The group ® contains as subgroups, the group % of enlarge-
ments, the group ft of isomctries and the group X of transla-
tions.
Problem 101, Discuss the construction considered in Problem
100 for special positions of the points A, A' and B, B\
SIMILARITIES 105
Problem 102. Given are two spiral similarities
W&Sift&d and W 2 (S 2 ,B 2 ,n 2 ).
Determine the centre, angle of rotation, and scale factor of the
spiral similarity W = W 2 W±.
To solve this problem we construct two pairs of correspond-
ing points for W; the problem then reduces to Problem 100.
The construction is particularly simple if we select A to be
the point S x and B to be the point which is mapped onto S 2 by
the transformation W t . We have: 8 =* 0i + 2 i /* = /'i^-
Problem 103. Construct a cyclic quadrilateral given the lengths
of the four sides.
Consider a spiral similarity
with centre A which maps B onto
D. The image C of C lies on
the line through C and D, since
opposite angles of the quad-
rangle are supplementary (see
Figure 78). For this spiral simi-
larity we have 8 = a and n = dja.
If we start the construction by
drawing side c, we can first find
C, using b' = b.dja.
We now have two geometric
loci for A; we have AD = d,
and also AC: AC = d:a (Apoi-
lonius's circle). Ft'g7&
Problem 104. A B is a given line segment and g an arbitrary line
through A. If we map AB by all possible spiral similarities with
a given centre S which leave the image of A on the line g, show
that the geometric locus of B' is a line s. What relation exists
between the two lines g and 5?
Consider the different images of AB and interpret the corres-
pondence between starting point and end-point of these line
segments as a mapping.
106
TRANSFORMATION GEOMETRY
OPPOSITE SIMILARITIES
S
\
\
\
~;=^A'
4*
S Fig 79
As the simplest example of an opposite similarity we consider
the so-called stretch-reflection. We obtain this type of transla-
tion by combining a reflection M s with an enlargement E which
has its centre on s. As can be deduced immediately from Figure
79, the stretch-reflection Z does not depend upon the order in
which the reflection and the enlargement are carried out:
Z = EM, m M S E.
The class of opposite similarities is also dominated by a
single type of transformation:
Theorem 17. Every opposite similarity is a stretch-reflection.
The proof is similar to the corresponding one for direct
similarities. We refer again to Figure 75 and the transforma-
tions defined there. If, in addition, we denote the line through
A' and B' by r, we can represent fi* by the following product:
CI* - EM F M„M P = EK*.
K* is an opposite isometry. For the further reduction of Ci* we
consider Figure 80 which illustrates the position of the three
lines p, q, r and the centre S.
SIMILARITIES
107
Fig 30
From this figure we deduce
£1* = EM r M q M p m EM ri {M q2 M p ) m M r EH L .
The transformations M rj and E commute since 5 lies on r 2 ,
EH L m E* is again an enlargement with centre S* on q z . If we
now introduce the perpendicular 5 to r 2 through S*, we obtain
Cl* = M, 2 EH L = M f E* m (M ri MXM s E*) - H M E*M S .
Here we have used the fact that E* and M s commute since 5*
lies on s. Ef = H M E* is an enlargement with centre S* on s.
We have now obtained the following representation for CI*'.
Cl* = EW S .
This is a stretch-reflection, since S* lies on s.
The essential elements of a stretch-reflection are the axis of
reflection s t the centre of enlargement 5 on s, and the scale
factor fi.
Problem 105. The two pairs of points A, A' and B, B' are given.
Describe the stretch-reflection Z which maps A onto A' and B
onto B'.
The scale factor n is given by A'B'jAB. It can be seen from
Figure 79 that the axis s divides the line segment A A' in the
ration 1 : ft. Therefore we can immediately construct two points
of 5.
108
TRANSFORMATION GEOMETRY
Problem 106. Show that the set of all spiral similarities and
stretch-reflections under the operation of combination of
transformations forms a group ©*.
©* is the group of all similarities. The geometry of this group,
in the sense of Klein, is called the geometry of similarity.
To this geometry belong all those geometric properties of
figures which are invariants with respect to ©*. Examples are
the orthocentre of a triarjgle, the mediator as the geometric
locus of all points equidistant from two given points, the circle
as the geometric locus of all points equidistant from a given
point. We can say that circles form an invariant class of curves
with respect to ©*. Pythagoras's Theorem also belongs to the
geometry of similarity, for if we have a 1 +b 2 = c 2 for a triangle,
the same is true for the image triangle.
Problem 107. The spiral similarities form a subgroup S of <3*.
Why do the stretch-reflections fail to form a subgroup ?
Problem 108. Determine the group <5 f of all similarities leaving
the pattern in Figure 70 invariant. List also some subgroups of
Problem 109. What kind of transformation is *¥K<¥- ■ if Ke ft
and^feS*?
What do we obtain when K is :
(a) a reflection,
(b) a half-turn,
(c) a translation,
(d) a rotation,
(e) a glide-reflection ?
Definition, If a geometric figure F can be mapped onto a figure
F' by a transformation *P e S*, then we say that F and F' are
similar figures.
Problem 110. Show that the relation of similarity defined above
is an equivalence relation.
Problem 111. A triangle ABC is given. Corresponding to each
SIMILARITIES
109
side construct a vector which is at right-angles to the side, is
twice as long as the side, and which points outwards from the
triangle. Prove that the vector sum of the three vectors is zero.
Problem 112. A, B, C, D are the vertices of a square (described
in an anticlockwise direction). Consider the three spiral
similarities
WM> 45°, V2).
W 2 (B, 45°, 72),
W 3 {C, 90°, i).
What transformation is W i W % W x 1
Problem 113. What kind of transformation is *F 2 if *F is a
stretch-reflection?
AFFINE TRANSFORMATIONS
The transformations we have discussed so far were really
elementary geometry in a new guise. Now we come to the
first example of a mapping which does not belong to the frame-
work of elementary geometry. We enter at the same time a field
of geometry of rather more recent origin; affine transformations
were first introduced in 1748 by Leonhard Euler (1707-1783)
in his analysis of the infinite.*
Let us select a line s in the plane and a direction given by the
angle a it makes with s. We also choose a real number p,
positive or negative but not zero.
Wc can now define a mapping of the plane by the following
law of correspondence:
The lines through pairs of corresponding points P, P' are
parallel to the given direction.
* Euler L. : Introductio in analysin infinitorum. Tomus secundus. Cap
XVIII, De similitudine et affinitate linearum curvarum. Opera omnia 1/9.
UO
AFFINE TRANSFORMATIONS 111
For every pair of points P, P' we have
P'P* _
where P* is the intersection of the line PP' with s.
This transformation is called a perspective affinity^; s is the
axis of affinity, the given direction is called the direction of
affinity, and p. is called the scale factor of the affinity.
When we wish to describe the transformation fully we shall
use the symbol <b(s, «, p).
An important special case of a perspective affinity is reflection
in a line, for
M E = ®(s, 90", - 1).
Perspective affinities for which a — 90" are called normal
affinities.
PROPERTIES OF PERSPECTIVE AFFINITIES
If we are given a point P and its image P' under a perspective
affinity with axis s, then further pairs of corresponding points
can easily be constructed. We can use the construction shown
in Figure 81 to ensure that
QQ*
P'P*
PP*
- v-
If Q moves along the line g, then Q' moves along g' (in our
construction only the parallel to the direction of affinity moves).
Hence, g' is the image of g under the affinity.
If A is parallel to j, then the image points of h lie on a parallel
line h'. This is a consequence of the fact that if P moves along
the line h, then the distance PP* remains constant; hence
P'P* is constant, too.
1. A perspective affinity is a line-preserving transformation.
Corresponding lines intersect on the axis of affinity.
The line j is a fixed line; in fact, all points off are fixed points.
The lines parallel to the direction of affinity are also fixed lines,
but, unlike s, they are not pointwise fixed.
t Or axial stretching.
112
TRANSFORMATION GEOMETRY
The verification of the following properties offers no parti-
cular difficulties' accordingly no detailed explanation is neces-
sary.
2. A perspective affinity maps parallel lines onto parallel lines.
3. The transformation preserves ratios of division; the ratios of
distances of three points on a line g are equal to the corres-
ponding ratios for the image points on g\
4. The ratio of the area of a polygon to its image under a
perspective affinity is 1 : ft*
5. A perspective affinity is a one-one mapping. The inverse
affinity has the same axis and direction but has scale factor
If a perspective affinity is to preserve the (non-oriented) area
we must have ft = ±1. If ft = - 1 we obtain the so-called
oblique reflections (see Figure 82(a)), which we can see to be a
generalization of reflection in a line. The transformations with
ft = 1 obviously include the identity /; however, this is not the
only affinity with this property. If we note that the scale factor
can also be expressed in terms of the oriented distances of
corresponding points A and A' from s, we deduce that if ft = I
these distances are equal. If * ^ /, then the direction of affinity
must be parallel to s. In this case we speak of a shear with the
axis s (see Figure 82(b)).
•We are here considering orientated areas. The ratio of the areas is
positive or negative according to whether the original and image polygons
have the same orientation or not.
AFFINE TRANSFORMATIONS
113
If « = 0, ft can no longer be chosen arbitrarily, ft is then
always + 1. For a shear or and /« are not independent, and as a
consequence the transformation is not completely determined
by s, <x and fi. A shear is a degenerate type of perspective
affinity.
As parallelism is invariant, a square is mapped by a perspec-
tive affinity onto a parallelogram. A given quadrangle can only
be mapped onto a square by a perspective affinity if its opposite
sides are parallel. The following problem shows that this condi-
tion is not only necessary but is also sufficient.
Fig. 83
Problem 114. Find a perspective affinity with given axis s which
maps a given parallelogram A BCD onto a square.
The image of a parallelogram under a perspective affinity is
always another parallelogram, since parallel lines have parallel
images. Hence, the parallelograms form an invariant class of
quadrangles with respect to the perspective affinities. The
image parallelogram is a square if the angle 0' between two
sides is 90° and the angle <p' between a side and a diagonal is
45°. The conditions
e -» 0' = 90° and <£->0' = 45°
define two geometric loci for A'. Their intersection is A'. Once
we have a pair of corresponding points we can, of course, easily
construct the transformation. The complete geometric loci have
114
TRANSFORMATION GEOMETRY
a second poiot of intersection which is the image of A' utider
reflection in s. If we ask for the transformation, the problem
has two solutions. The two squares obtained are, however,
congruent and we shall later prove this.
Among the numerous properties which characterize quad-
rangles, only those which characterize parallelograms are
invariant under perspective affinities. Examples are the proper-
ties that opposite sides are parallel, or that the diagonals bisect
each other. In general, a perspective affinity will not preserve
angles.
Problem 115. Show that a hexagon can be mapped by a pers-
pective affinity onto a regular hexagon if and only if its diagonals
are concurrent and pairs of opposite sides of the hexagon are
parallel to a diagonal.
The two conditions are certainly necessary, for they are satis-
fied by a regular hexagon and they are invariant under perspec-
tive affinities.
We can obtain a hexagon satisfying these conditions in the
AFFINE TRANSFORMATIONS
115
following way (see Figure 84). We start with the three diagonals
a, b, c intersecting in a point M. We choose a point i>, on a and
draw through it a line parallel to c; this determines P 2 on b.
Then draw a line parallel to a to obtain P 3 , etc. Finally we
obtain P 6 . The parallel to b through P 6 passes through P x and
completes the hexagon. This follows from the relation
P t P 2 = MPs = P A P S = MP 6 .
We can now prescribe the axis of affinity s in an arbitrary
manner.
If we determine the affinity in such a way that a, b, c are
mapped onto three lines intersecting each other at angles of
60°, then the image of P t P 2 . . . P 6 will be a regular hexagon,
since the six triangles which make up the image hexagon will be
equilateral.
Problem 116. Given a triangle ABC, a line s and a point S\
construct a perspective affinity having s as its axis and such that
S' is the centroid of the triangle A'B'C.
The position of the centroid of a triangle is determined by
certain ratios of line segments. Since such ratios are invariants
under perspective affinities, it follows that the image of the
centroid of ABC is the centroid of the image triangle A'B'C.
The notion of a centroid is invariant under perspective affinities.
We can construct the centroid S of the given triangle ABC.
S and S' are then corresponding points which determine the
transformation.
The centroid S of a quadrangle
ABCD can be found by two
different decompositions into
triangles (see Figure 85). This
combined construction is also
affine, that is, it is built up using
properties which are all invari-
ant under perspective affinities.
The affine image of the con-
struction coincides with the
corresponding construction for
the affine image of the quad-
rangle.
116
TRANSFORMATION GEOMETRY
Problem 117. Verify the con-
struction of F. Wittenbauer* for
the centroid of a quadrangle.
This construction is shown in
Figure 86 and is also invariant
under affine transformations.
The sides of the quadrangle are
divided into three equal parts
and a parallelogram is then ob-
tained by drawing lines through
pairs of points of trisection ad-
joining a vertex. The sides of this
parallelogram are parallel to
the diagonals of the given quad-
rangle and the diagonals of the parallelogram intersect in the
centroid of the quadrangle.
Wittenbauer's construction can be reduced to the construc-
tion shown in Figure 85 by purely geometric means, that is,
without calculation.
Problem 118. Give a construction for finding the centroid of a
pentagon.
Problem 119. Given are the axis of a perspective affinity and a
pair of corresponding points A and A'. Construct two orthogonal
lines (i and v which intersect at A and which have orthogonal
images.
The problem is solved by constructing a circle which passes
through A and A' and has its centre M on s. M lies on the media-
tor m of the line segment A A'. Figure 87 is based on a perspec-
tive affinity having a scale factor /*>0.
Since a perspective affinity preserves parallelism, there are
two orthogonal directions which are always mapped onto
orthogonal directions. We always obtain this same pair of
directions whatever pair of corresponding points we use. We
call these two directions the invariant right-angle pair of the
transformation.
* Sec Strubccker, K., Einfiihriirig in die hshere Mathematik. Vol. ], p.
403, Munich, 1956.
AFFINE TRANSFORMATIONS
117
Fig 87
This construction runs into difficulties if the direction of
affinity is at right-angles to the axis, that is, for normal affinities.
Consider first the case p 96 — 1, that is, the transformation is
not a reflection. Then m is parallel to s but different from s (see
Figure 88). The circle through A A' now becomes a straight tine
through these points. The invariant right-angle pair exists in
this case too ; one direction is parallel to, the other orthogonal
to, the axis of affinity.
For a reflection (ji = -I) m and s coincide; the point of
intersection is, therefore, indeterminate (see Figure 89). Every
point of s can be taken as the centre of a circle passing through
A and A'. Hence, every right-angle is invariant. The result
,m
A>
Fig. 89
Fig. $3
118
TRANSFORMATION GEOMETRY
agrees with the fact that a reflection is an isometry.
We can sum this up as a further property of our transforma-
tions:
6. Every perspective affinity which is not a reflection has
exactly one invariant right-angle pair. As far as this property
is concerned reflections are degenerate cases; under reflec-
tions every right-angle pair is invariant.
Problem 120. A perspective affinity is given by its axis s and a
pair of corresponding points A and A'. Construct an invariant
60°-pair of lines which meet at A.
We first consider the case ft>0, that is, A and A' lie on the
same side of s. Then our problem is to construct a circle through
A and A' such that the axis s cuts off an arc which subtends an
angle of 120° (or 240°) at the centre of the circle. This problem
has already been solved in Problem 77.
If /*<0, that is, if A and A' lie on opposite sides of s, we first
reflect A' in s. This leads to a point A" and reduces the problem
to the case /t>0 already considered. The problem always has
two solutions.
In the solution for /i<0 there appears, besides the perspec-
tive affinity <J>, the reflection A/ s . In order to find the invariant
AFF1NE TRANSFORMATIONS
119
60 c -pair for the transformation C>, we first determine the
corresponding pair for the mapping M&. This is again a per-
spective affinity with axis s.
The importance of perspective affinities lies in their connec-
tion with the parallel projection. For this reason construction
problems on perspective affinities can be found in many collec-
tions of problems in descriptive geometry. Therefore we
restrict ourselves here to considering certain problems which
bring out new aspects.
THE GROUP PROPERTIES OF PERSPECTIVE AFFINITIES
The remark at the end of Problem 120 leads us to the conjec-
ture that the perspective affinities with a given axis form a
group. In order to check this we shall use methods of analytic
geometry.
We consider first a
perspective affinity <b(s,
at, fi) and set up a rect-
angular coordinate sys-
tem as shown in Figure
91. The .v-axis is taken
to coincide with the axis
of affinity 5. Our trans-
formation 0, which
maps A onto A', is then
characterized by the
equations:
x' = x+(ft— 1) cot a y,
/ = 99*
However, we have seen that shears cannot be characterized
by the parameters a and u. In order to find a standardized
representation for all perspective affinities we have to look for
more suitable parameters.
Since the transformation O preserves parallelism, the angle
between a, the perpendicular from A to s, and its image line a'
is independent of the choice of A. Corresponding to every
perspective affinity 3> there is a well-defined angle a. Now we
have
Fig 9}
120
or
TRANSFORMATION GEOMETRY
x—x' 1— U
tan a m ■ > — = £—
y ft tan a
Qi— 1) cot a m — ^ tan <r.
We now replace a in our transformation equations by the
parameter a. The equations can then be written :
x' = x— ft tan a y,
/ = W>
and it is easy to see that these equations are also valid for
shears. Tan a always has a finite value.*
Problem 121. A perspective affinity * has axis s, angle
a = 45°, and scale factor ft = J. Construct a pair of correspond-
ing points.
Problem 122. Calculate the angle a' for the inverse transforma-
tion <E» -1 {G> is defined as in Problem 121).
We now have the necessary analytic tools to prove the
following theorem:
Theorem 18. The perspective affinities with a given axis s form
a group Qj.
Let *i and <t> 2 be two perspective affinities having the same
axis 5. We use the notation (see Figure 92)
A(x,y)— >A'(x',y')-^ A"(x",y").
Then the equations of the transformations are
= x'~ft 2 tanff 2 /,
ft W = x-^ tan &, ,v, jx" = *>-§
%' - ft* %" = ft*
The geometric composition of <t>! and <& 2 so as to form the
new transformation * = <t> 2 <&[ corresponds to the algebraic
* Note that n and a are directed angles.
AFFINE TRANSFORMATIONS 121
substitution of the equations for *, into those for * 2 . We
obtain
jc" = x-ti x tan ff, y-Pifo tan a 2 y
= x—(jii tan ffi +/i,/i 2 tan a 2 )y,
y" - fhfhy-
These equations, too, have the structure of transformation
equations for a perspective affinity with axis s. If we write
x" = x-t* tan a y,
y" = ny<
then we obtain by comparison,
fi - ftft.
fi tan a = /*i tan ffi + /i,jt 2 tan a 2 .
This implies the following rules of composition for the scale
factors and angles:
* * * I * m ft***
' * I tan tr = l//i 2 tan ffj +tan cr 2 .
Hence, the mapping obtained by combining two perspective
affinities with axis s is again a perspective affinity with axis 5.
The verification of the remaining group postulates is trivial.
Corresponding to every line 5 there exists a group Q,. This
fact is expressed by using the suffix s in the symbol for the
group.
-A
A
%^ \
/ L \ .\J&z
Fig 92
122
TRANSFORMATION GEOMETRY
The group Si s is not commutative. This can be seen immedi-
ately from the rule for obtaining the angle of the product
transformation, since the formula for a is not symmetric in the
pairs 0*i, tan a % ) and {fi 2 , tan a 2 )-
Problem 123. Show that in the group Si, we have the following
rule for obtaining the direction of affinity of the product
transformation :
tan a = (1-^ifo) tan ct t tan« 2
(1 -a*!) tan a 2 +Ml -j»a5 tan %*
Problem 124. Prove the theorem of Menelaus using transforma-
tions belonging to the group Q, (cf. Problem 87),
We are now in a position to settle a question which arose in
connection with Problem 114. Then we constructed an affinity
with a given axis s which mapped a parallelogram onto a square.
We found that there was a second transformation having this
property and that
a> 2 = Af,*,.
But from this it follows at once that the two squares obtained
are congruent (and are symmetrically placed with respect to the
axis).
Problem 125, Show that two transformations * lf <t> 2 c Q s
commute if, and only if, their directions of affinity coincide
(«i = «a)-
From the rules of composition we found
1 1
* 2 *j = *,*2<»— tan 0-,+tan s 2 = — tan <r 2 + tan a t .
A*2 Hi
This can be written
r^; tan °> = t~ ta *>■
But the expression on the left-hand side represents cot «j and
that on the right-hand side cot a 3 , hence, ot, = a 2 ,
We can easily find subgroups of O r We note that s is a fixed
AFFINE TRANSFORMATIONS
123
line for all * e £),. By demanding further invariants we obtain
subgroups of £5j.
If we demand that a certain direction should be an invariant,
we obtain the subgroup O jW of all the transformations having
the same direction of affinity (given by the angle a). It follows
from Problem 123 that C 1(a) is a commutative group; the
composition of mappings in the group being given by the
relation p = Hiit 2 obtained above.
Another subgroup Q S '<=C S can be obtained by demanding
the invariance of the measure of area. O,' contains the perspec-
tive affinities with ft ■» ±1, that is, shears and oblique reflec-
tions. The shears alone form a group Q/'eQ',.
Problem 126. Show that in the group Si" the composition of
transformations can be described by the relation
tan g = tan ^H-tan <r 2 .
Illustrate this result diagrammatically.
.4
Problem 127. Interpret the two constructions shown in Figure
93, in which the two image figures have the same area as the
original figures, in terms of mappings belonging to O,.
124
TRANSFORMATION GEOMETRY
Problem 128. Show that shears having the same axis s and such
that tan a = na where n is an integer and a a constant, form a
group. Try to give a geometric figure which is invariant with
respect to this group (cf. Figure 70).
The product of perspective affinities having different axes is,
in general, not a perspective affinity. For example, the product
of two reflections is a rotation or a translation. These are
mappings having either one fixed point or none at all and,
therefore, cannot be perspective affinities.
Problem 129. Two perspective affinities <&i and <D 2 have axes
which intersect in a point S, The direction of affinity of each
mapping is parallel to the axis of the other, and we have
ft, = p 2 = ft. Show that ^O^ is the enlargement E(S,n) and
that the two transformations <b x and <b z commute.
Fig 94
Figure 94 illustrates the situation. The problem reveals the
fact that every enlargement E can be represented as a product
of two perspective affinities.
Problem 130. Two perspective affinities 4>, and 2 have axes
jj and s 2 which intersect in a point S. Again we assume that the
direction of affinity of each mapping is parallel to the axis of
the other, but now we suppose that //,/x 2 = 1.
(a) Show that <tf 2 Q> l is a transformation which preserves lines
and area, and that S is a fixed point of the transformation.
(b) Let A be a point on s t and B a point on s 2 . Show that the
AFF1NE TRANSFORMATIONS
125
lines through the pairs A, B' (= <J> 2 *i(£)) and B, A'
(= 9 2 9x(A)) are parallel.
(c) Show that <I> 2 <t>, = 4>i1> 2 .
Problem 131. <t>! and <P 2 are two given perspective affinities
satisfying the conditions of Problem 130. Setting fi t = A, we
see that ft 2 = \fX, and that we can write
^O, = *(A).
(a) Show that the transformations 4>(A) (for all X # 0) with
given axes 5! and s 2 form a commutative group &(s u s 2 ).
(b) Translate the group relation
W.) = WJMd
into geometric language. (Use the result of Problem 130(b) — the
Pappus-Pascal configuration.)
Problem 132. Show that hyperbolae with asymptotes 8%
and s 2 are invariant under transformations of the group
If *i and <D 2 are two arbitrary perspective affinities, then the
transformation * 2 *i * s obviously a mapping which preserves
lines, parallelism and ratios of division. These properties of
<!>! and * 2 are not lost by composition. We have met other
transformations with these properties, for example, the ele-
ments of the groups ft and S*.
Definition. Every finite product of perspective affinities is called
a general affine transformation or, for short, an affinity.
It follows from the definition that affinities are transforma-
tions which preserve lines, parallelism and the ratios of division.
They form a group Q, the full affine group. The groups £Jj are
subgroups of Q. Moreover, ft and S* are also subgroups of Q,
since we have seen that the reflections, which generate ft, are
particular perspective affinities, and Problem 129 showed that
t This property is used in the geometric introduction of the natural
logarithm . See, for example, Van der Waerden B. L., Die Einfiihrung des
Logarithmus im Schulumerricht. El. Math. 12, 1957.
126
TRANSFORMATION GEOMETRY
every enlargement could be obtained as the product of two
perspective affinities.
Problem 133. Given are two triangles ABC and A'B'C Show
that there exist a perspective affinity <t» and a spiral similarity
Q such that the transformation £1$ maps the triangle ABC onto
the triangle A'B'C
Take, for example, the spiral similarity which maps the line
segment AB onto the segment A'B'. Then it is easy to find a
perspective affinity * such that £1* has the required property.
The reader will easily work this out with the aid of a simple
drawing.
The transformation QO which we have just constructed is an
affinity. Hence, there always exists an affinity mapping a given
triangle ABC onto a second given triangle A'B'C, and there
exists only one such affinity as we shall proceed to show. If we
want to construct the image of an arbitrary point P, we can do
this in the following way. We draw lines through P parallel to
AC and AB. Let these lines intersect AB and AC in the points
P t and P 2 respectively. Since ratios of division are invariant
we can immediately construct the images i>,' and P 2 ' on the
sides A'B' and A'C of the image triangle. If we then draw
lines parallel to A'B' and A'C through these points, we find
P', the image point of P. Hence, the image of any point, and
therefore the affinity, is uniquely determined by the triangles
ABC and A'B'C. There can be only one such affinity. The
decomposition into a spiral similarity and a perspective affinity
is, however, not unique.
The group Q has finite subgroups as well. If we ask, for
example, for the affine transformations which leave a given
triangle ABC invariant, then we see that any such transforma-
tion produces a permutation of the three vertices. Since there
are only six permutations, it follows that there are only six
affine transformations of the triangle onto itself, and that these
six elements form a group. We shall follow up this example by
looking in more detail at the affinities which map a given
parallelogram A BCD onto itself. Every such affinity again
results in a permutation of the vertices. Since the affinity is
determined by the images A', B\ C of the points A, B, C, we
AFFINE TRANSFORMATIONS
127
can characterize the affinity by giving the positions of A', B',
C on the parallelo-
gram. This choice is -^o \ c,
not arbitrary, for D has
to be mapped onto a
vertex as well. This will
only happen if A', B', C
are (in this order) ad-
joining vertices of the
parallelogram. There
are eight possibilities
for this. Four of these preserve the orientation, and four re-
verse the orientation of the parallelogram. There are, therefore,
eight affinities which leave the parallelogram invariant. They
form a group Q 8 and the eight mappings can be described as
follows (see Figure 95):
/ identity transformation
"t>, affinity moving the vertices round one place in an anti-
clockwise direction
<D 2 affinity moving the vertices round two places in an anti-
clockwise direction
4> 3 affinity moving the vertices round three places in an anti-
clockwise direction
$/ oblique reflection with axis s,
<I> 2 ' oblique reflection with axis s z
d> 3 ' oblique reflection with axis s 3
4> 4 ' oblique reflection with axis s 4 .
The group table for these transformations is
/
*!
<K 2
* 3
•l'
* S '
*3'
<V
/
/
*.
*2
<I»3
*l'
*,'
*>'
<V
*1
®i
*2
<»3
/
*3'
<v
<J> 2 '
*i'
®2
*2
a> 3
/
<&1
4>2
<v
<V
<V
*3
®3
/
*1
*2
**'
<v
*,'
0> 2 '
*/
<v
<V
**'
*3'
/
*2
*3
*1
<V
0,'
*s'
«v
<V
*2
/
*1
*3
<V
<v
*,'
<v
*,'
•l
*3
/
*2
**'
*/
*2
4» 3 '
<v
*3
*i
*2
/
128
TRANSFORMATION GEOMETRY
If we compare this table with the group table for the iso-
metrics which map a given square onto itself, we see that the
two groups have the same structure although they originated
from different problems. We can establish a one-one corres-
pondence between the elements of the group <5 8 for the square
and the elements of S2 8 , in such a way that the product of two
elements of © 8 corresponds to the product of corresponding
elements in Q 8 . Two groups for which such a one- one corres-
pondence can be defined are said to be isomorphic.
In Problem 114 we showed that every parallelogram can be
mapped onto a square by some affinity. If we denote this
affinity by T, then we have the following relations between the
elements of <5 8 and Q 8 :
Rj = "¥$^-1
and we have, for example,
Mj = T*/^ 1 ,
M 3 R t =(¥it> 3 'T- 1 )(V<Di4'- 1 )
This describes the isomorphism between the two groups
geometrically. The two problems leading to the groups <5 8 and
Q 8 were not, therefore, basically different. The transformations
4>i, * 2 , $3 are rotations deformed by an affinity, and *,',
®i, GV. */ are reflections which have been deformed.
Another interpretation of the isomorphism springs from the
fact that the groups © 8 and Q 8 describe the same group of
permutations of the vertices A, B, C, D.
Problem 134. Show that the perspective affinities with the same
direction of affinity form a group.
Problem 135. Show that if an affinity has a pointwise fixed line
s, then it is a perspective affinity with axis s.
Let A, A' be a pair of corresponding points for the given
affinity <J> and suppose that A does not lie on s. Making use of
the fact that affinities preserve ratios of division, we can con-
struct the image point P' of any point P from s, A and A'. This
image is unique; hence it follows that there is only one affine
transformation which leaves all points of s fixed and maps A
AFFINE TRANSFORMATIONS
129
onto A', But the perspective affinity with axis 5 and the pair of
corresponding points A, A' is such an affinity.
Fig 9$
Problem 136. Show that every perspective affinity <& can be
represented as a product of a spiral similarity Q. and a normal
affinity *„.
Let * be a perspective affinity with the axis s, and let A and
A' be a pair of corresponding points under 4>. Construct the
invariant right-angle pair at A and A' (see Figure 96). Now we
consider a spiral similarity £2 with centre S, angle and scale
factor ft = SA'fSA.
The images of A, g and h under the transformation Q are
denoted by A", g" and h". By the definition of fl we have
A' = A", g' = g" and h' = h". For the image T" of T under
fi, we have
A'T" m nAT.
Now let <&„ be the normal affinity with axis s a = g' = g",
which maps T" onto T, Then the transformation 0„£1 has both
S and Tas fixed points. Since ratios are invariant it follows that
every point on the line j through S and T is also a fixed point.
Hence, using the result of Problem 135, $„Q is a perspective
affinity with axis s. However, it maps A onto A' and so it must
i
130
TRANSFORMATION GEOMETRY
be the given affinity fl>. Hence, <& = d>„£i which shows that any
perspective affinity can be decomposed into a spiral similarity
and a normal affinity, f
AFFINE GEOMETRY
Affine geometry, in the sense of Klein, is the study of invari-
ants of the group G. Hence, all those properties of figures which
are preserved under affine transformations belong to affine
geometry.
The group @*, generated by reflections and enlargements, is
a subgroup of Q. This means that every property belonging to
affine geometry also belongs to the geometry of similarities.
The reverse is not true; a property of similarity geometry is not
necessarily an affine invariant. For example, we mentioned
before that the relation
a 2 + b 2 = c 2 ,
for the sides of a triangle, is an invariant under similarities. An
affine transformation will not, in general, preserve right-angles.
Since this relation characterizes right-angles it will not, there-
fore, be an invariant under Q. Affine geometry embraces only
a subset of the properties which belong to similarity geometry.
The theorem of Pythagoras is not one of these; the group of
similarities forms an extreme limit for this particular theorem.
However, ratios of areas are conserved by affine transforma-
tions. To measure an area means to find out how many times a
certain unit area is contained in it. The measure of area is,
therefore, based on a relation of the form
F, = pF 2 .
From the result of Problem 133 we conclude immediately
that this relation is preserved by affine transformations. For the
affine images we have again
Ft = pjy.
An example of a property which is affine invariant is that the
diagonals of a parallelogram divide the figure into four triangles
of equal area.
+ This decomposition is not unique. Wc could, for example, start wish a
spiral similarity with centre 7".
AFFINE TRANSFORMATIONS
131
The notion of a vector also belongs to affine geometry. This
follows from the fact that parallel line segments having equal
length are mapped onto image segments which are again paral-
lel and of equal length. Every relation between vectors is an
affine invariant; for example, if
v m a+3A,
then die same relation holds for the image vectors.
Every group defines an equivalence relation between those
figures that can be mapped onto each other by transformations
of the group. For example, all triangles are equivalent with
respect to S, for an affine transformation can be found which
maps any given triangle onto any other triangle. There is only
one class of triangles in affine geometry. In similarity geometry,
however, such triangles as the isosceles right-angled triangles
form a class by themselves. Parallelograms also form one class
in affine geometry, but not trapezia. Two trapezia only belong
to the same equivalence class if the ratio of the lengths of the
parallel sides is the same. Only then can one be obtained from
the other by an affine transformation.
Problem 137. What conditions must be satisfied by a hexagon
if it is to belong to the same equivalence class as a given regular
hexagon :
(a) with respect to the group ®,
(b) with respect to the group Q?
Problem 138. Show that two quadrangles belong to the same
equivalence class with respect to Q if the diagonals in both
figures divide each other in the same ratio.
Problem 139. Show that areas of corresponding polygons under
a general affine transformation are in a fixed ratio ft.
10
THE AFFINE GEOMETRY OF THE
ELLIPSE
We start with the following definition :
Definition. The image of a circle under a perspective affinity is
called an ellipse.
Since the perspective affinity preserves ratios, every diameter
of the circle T is mapped onto a diameter of the ellipse V. This
fact can be expressed in the language of transformation
geometry by writing
<&H„<!>- 1 = H,
M'
Corresponding to orthogonal diameters of the circle we ob-
tain what are called conjugate diameters of the ellipse (see
Fig $7
132
THE AFFINE GEOMETRY OF THE ELLIPSE
133
Figure 97). The property of orthogonality is not an affine
invariant, but orthogonal diameters of a circle can be charac-
terized by a property which is invariant under affine transforma-
tions. This property is that the tangents at the end-points of one
diameter are parallel to the other diameter. Conjugate dia-
meters of an ellipse will, therefore, also be characterized by the
fact that the tangents at the end-points of one diameter will be
parallel to the other diameter. Among these pairs of conjugate
diameters there will be one pair which is orthogonal; this pair
defines the axes of the ellipse. The axes are, therefore, related to
the invariant right-angle pair of the affinity.
Problem 140. Construct the axes of an ellipse which is the image
of a circle T under the perspective affinity <t>.
Problem 141. Show that every ellipse is the image under a
normal affinity of the two circles on its axes.
We imagine that the given ellipse is the image of a circle T
under the perspective affinity $. If we construct the invariant
right-angle pair for at M and M' , then we can reduce the
problem to the decomposition of $ into a spiral similarity fl
and a normal affinity <S> ni as described in Problem 136. We
denote the two normal affinities by ® nt and O rt2 . If a and b are
the lengths of the semi-axes of the ellipse, we have
vjsu 90 s , ±^j and <S>Js 2i 90°, ±f\.
The sign of p t and fi 2 depends on the sign of the scale factor
of the transformation O.
Problem 142. Show that $ n 2 _1< Diit ' s an enlargement with
centre M', and deduce the method shown in Figure 98 for
constructing points on an ellipse from points of the circles on
its axes.
Draw in the tangents to the curves at the three related points
P u P 2 &n6P\
We now wish to show that every ellipse T' can be obtained as
a plane section of a circular cylinder. According to Problem 141
there exists a normal affinity <t>„ 2 which maps the circle T 2 on
the minor axis onto I"'; the scale factor of the affinity being
134
TRANSFORMATION GEOMETRY
Ql
/tf
V
AN
I s i I «'
s 2
— -^ Figs
alb. We now imagine the ellipse V turned out of its plane. We
choose s 2 , the axis of <&„ z , as the axis of rotation and make the
angle of rotation 4> ~ cos -1 6/dr. The points P 2 and P', which
were corresponding points under O n2 , will, after this rotation,
lie on a line orthogonal to the plane of the circle F 2 . Hence, the
ellipse now lies on the cylinder over the circle F 2 (see Figure
99).
Fig 99
Problem 143. Show that the ellipse is the locus of all points
whose distances from two fixed points F t and F 2 have a constant
sum.
THE AFFINE GEOMETRY OF THE ELLIPSE
135
We use the fact that every ellipse can be obtained as the
intersection of a plane e with a circular cylinder.
Imagine that we take two spheres of radius b and push them
into the cylinder from opposite ends until they touch the plane
e. Let the two points of contact be F, and F 2 . It is then easy to
show that for all points P' on the ellipse the sum P'F i +P'F 2
has a constant value. The generating line of the cylinder
through P' touches the two spheres in points A and B. We now
use the fact that all tangents from a point P' to a sphere have the
same length. Hence, we obtain
P'F l +P'F 1 = P'A+P'B m AB.
AB is the segment of the generating line through P' which is
bounded by the two circles of contact between the spheres and
the cylinder. It can be seen immediately that AB does not
depend upon the position of the point P' on the ellipse.*
The two points F t and F z are known as the foci of the ellipse.
Problem 143 introduced an important metric property of
the ellipse. Here, however, we are mainly interested in the affine
properties of ellipses and the following theorem takes us back
once more to affine geometry.
Theorem 19. A pair of conjugate diameters determines exactly
one ellipse.
The content of this theorem is not trivial since a pair of conju-
gate diameters can be mapped onto orthogonal line segments
having equal length, by infinitely many affinities. It is not
immediately obvious that the inverse transformations will
always map the circle onto the same ellipse.
We prove this by giving a construction for an ellipse starting
with a pair of conjugate diameters. It is necessary that this
construction is an affine invariant, that is, is built up from
affine invariant operations such as drawing parallels, or dividing
line segments in given ratios. We do not allow such operations
as drawing right-angles.
Let AB be a diameter of the circle F (see Figure 100), C a
* This method of proof is due to G. P. Dandelin (1794-1 847). The two
spheres appearing in the proof are sometimes called Dandeiin's spheres.
136
TRANSFORMATION GEOMETRY
point on F, and q an arbitrary perpendicular to AB. Since C lies
on the circle, LACB = 90°. We now extend AC until it meets
q and we letter this point T. His the orthocentre of the triangle
ABT. The line AP is the third altitude and is, therefore, perpen-
dicular to BT. We conclude that P, the point of intersection of
the lines BT and AH, lies on T. If we let q move (still, however,
remaining perpendicular to AB), then P will describe the whole
circle. Hence, Figure 100 shows a construction which allows us
to find any number of points on a circle without using com-
passes, provided a diameter and one other point of the circle
are given. Consequently we can solve the following problem
without using compasses :
Problem 144. Given two mutually orthogonal diameters AB
and CD of a circle P, construct the point of intersection of T
with a line g through B.
The solution is given in Figure 101(a).
Apart from drawing the right-angle between q and AB all
the steps in the construction are affine. We can, however,
interpret the construction completely in affine invariant terms,
if we interpret q not as the perpendicular to AB, but as a paral-
lel co the tangents to the circle at A and B (that is, q is the con-
jugate direction to AB). It should be mentioned that P can be
obtained directly by drawing the perpendicular to g through A.
THE AFFINE GEOMETRY OF THE ELLIPSE 137
Fig 107
However, the affine nature of the construction is lost by this
short cut.
An affine translation changes Figure 101(a) into Figure
101(b).
We can now base the proof of our uniqueness theorem on
Figure 101, If a given pair of conjugate diameters A'B' and
CD' is mapped by a perspective affinity onto an orthogonal
pair AB, CD which have the same length (the position of the
axis of affinity is irrelevant), then the inverse mapping applied
to the circle construction shown in Figure 101(a) always pro-
duces the same image. The reason for this is that all the points
and lines constructed in Figure 101(b) depend only on the given
pair of conjugate diameters. The position of P' on g' is, there-
fore, uniquely determined. Jf we now rotate g' round the point
B', then P' describes a well-defined curve — the ellipse with the
given conjugate diameters.*
Problem 145. The mid-point of the line segment HT in Figure
101(a) lies on the tangents to the circle F through C and P. Use
this property to obtain a construction for drawing tangents to
an ellipse.
* The remarkable configuration of Figure 100, on which this proof is
based, is given in Volume 3 of Enzykfopadie der EkmetUarmathematik by
Weber-Wellstein. It is a special case of Pascal's Theorem.
138
TRANSFORMATION GEOMETRY
Fig 102
Problem 146. An ellipse is given by means of a pair of conjugate
diameters. Determine the points of intersection of the ellipse
with an arbitrary line g.
We mention this simple problem on ellipses because its solu-
tion introduces a new means of using transformation geometry.
The given ellipse can be mapped onto a circle by an affinity.
The simplest mapping to use is one which leaves one of the
conjugate diameters invariant. That is, we consider a perspective
affinity <f> which has one of the two diameters as its axis of
affinity. The line g is mapped onto g' by <J>. Now the problem
has been transformed into one concerning a circle. We have to
find the points of intersection of g' with the image circle and
then construct the images of these two points under the map-
ping O -1 (see Figure 102).
In this problem we meet an idea which is frequently used in
higher mathematics. A given problem is reduced by means of a
suitable transformation to a simpler case for which the solution
is already known or at least obtainable more easily. The solu-
tion of the general case is then obtained by considering the
image of the simpler solution under the inverse transformation.
This is an extremely useful method of solution.
The following problem can be solved in the same way.
Problem 147. Two ellipses are given whose major and minor
axes are parallel and in the same ratio. Find their points of
intersection.
THE AFFINE GEOMETRY OF THE ELLIPSE
139
An affinity which transforms one ellipse into a circle will also
map the second ellipse onto a circle. Hence, a suitable affinity
reduces the problem to that of finding the points of intersection
of two circles.
Problem 148. Construct an ellipse which has the given diameter
AB and passes through two given points P and Q.
An ellipse can be constructed if it can be represented as the
affine image of a circle. Hence, we have to find a perspective
affinity which maps the required ellipse onto a suitable circle.
Fig 103
We take the line AB to be the axis of affinity of our mapping.
In a circle every chord is orthogonal to the line joining its mid-
point to the centre of the circle. The equivalent property for an
ellipse is that a chord and the line joining its mid-point to the
centre of the ellipse define conjugate directions. Therefore we
have to choose the affinity in such a way that the angle 4> (see
Figure 103) is transformed into a right-angle. If we translate the
angle tf> to P we have, together with the circle with diameter
AB, two geometric loci fozP' (since P' lies on the circle with UV
as diameter). Hence, we obtain two solutions for P' — the
points P t ' and P 2 '. Correspondingly there are two affinities
140
TRANSFORMATION GEOMETRY
* t and <& 2 to be considered. We now want to show that both
affinities map P onto the same ellipse, that is, there is only one
ellipse with the required properties.
We note that P 2 is obtained from JY by the reflection M a .
For the inverse transformations we therefore have
<V'
= ar 1 */..
However, the reflection M s leaves V invariant, hence, the
image of T' is the same for <b\
has a unique solution.
- 1
and <t 2 • Thus the problem
Problem 149. Two ellipses e t and e 2 have a common diameter
AB. Given P on e t and Q on e 2i and the tangents at P and Q,
construct the common tangents to the two curves.
To solve this we map e t and e 2 by two perspective affinities
onto the circle e' with diameter AB:
e «-
<b v <D 2
Then <& 2 -1 <&i = <P is a perspective affinity which maps ei
onto e 2 . The direction of affinity of <S> gives the direction of the
common tangents of e v and e 2 . If we denote the axis of affinity
of * 1 and <S? 2 by s, then the transformations
also map e t and e 2 onto the circle e'. Since we have
this combination produces the same affine relation between
ei and e 2 . The four possible combinations of 4> lt 0/ and
*z» ®i produce only two different transformations; to each
of these corresponds two common tangents of e t and e 2 .
Problem 150. Show that all ellipses belong to the same equiva-
lence class with respect to SD.
BIBLIOGRAPHY
The bibliography of the German edition listed 16 books of
which the following are published in English.
klein, F., Elementary Mathematics from an Advanced Stand-
point (Vol. 2, Geometry), Macmillan, New York, 1939.
weyl, H., Symmetry t Princeton University Press, Princeton,
1952.
An O-level course based on transformation geometry can be
found in the textbooks of the S.M.P. In Book T the emphasis
is on construction problems involving a single transformation —
the transformations considered are the isometries (excluding
glide-reflection), enlargements and shears. Book T4 introduces
products of transformations and the glide-reflection. The group
properties of transformations are not developed but an account
is given of the description in matrix terms of transformations
having a fixed point. Similar work on transformation matrices
is contained in Matthews. The Teacher's Guide to Book T has
a valuable section on motion geometry which will certainly be of
great assistance to any teacher who wishes to try out this
approach to geometry. Further guidance on class-room
presentation is contained in Chapter 10 of the A.T.M. hand-
book (this also mentions some of the group applications).
Rather more advanced accounts of the subject can be found in
two Russian translations: Yaglom, in which only isometries
are considered, and Kutuzov. Coxeter in his excellent, compre-
hensive book gives a short but readable account of the various
groups of transformations.
A footnote on p. 43 tells the reader where information on the
description of patterns and figures in group theoretical terms
can be found. Terpstra's pamphlet is intended to supplement
Escher's collection of designs and these, together with the
books by Weyl and Wells, make fascinating reading. A read-
able and inexpensive approach to symmetry and plane patterns
is given by Bell and Fletcher. The work on cycles of half-turns
and reflections on p. 72 is also done by Thomsen (but this
141
142
BIBLIOGRAPHY
article is now of interest mainly from a historical point of view)
and a full scale axiomatic treatment of this group theoretic
approach to geometry can he found in Bachmann (German).
Readers to whom group theory is new should follow up one
or two of the ideas hinted at in this book. A more detailed
account of cosets (the name given to the sets appearing in the
decomposition of © 8 on p. 78) and a proof of Lagrange's
Theorem (here stated without proof) can be found in any book
on groups. One of the cheapest and most readable is Leder-
mann. Problem 109 has deep implications in group theory as
the reader will discover if he investigates normal and conjugate
subgroups. An exciting, but expensive, treatment of groups is
given by Papy.
a.t.m. handbook, Some Lessons in Mathematics, Cambridge
University Press, London, 1964.
bachmann, f., Aufbau der Geometrie aits dem Spiegehtngs-
begrijff, (Grundlehren der mathematischen Wissenschaften,
96), Springer, Berlin, 1959.
bell a. and fletcher T. J., Symmetry Groups, A.T.M.
Mathematics Teaching Pamphlet No. 12, 1964.
coxeter, H. s. m., Introduction to Geometry, New York,
London, Wiley, 1961.
escher, M. c, The Graphic work of M. C. Escher, Old bourne
Press, London, 1961.
scutuzov, B. B., Geometry, (Studies in Mathematics, 4),
School Mathematics Study Group, New Haven (Conn.)
1960.
ledermann, w., Introduction to the Theory of Finite Groups,
Oliver and Boyd, Edinburgh, 1949.
Matthews, G., Matrices (2 vols.), Arnold, London, 1964.
papy, g., Groups, Macmillan, London, 1964.
school mathematics project, Book T, Cambridge Uni-
versity Press, London, 1964.
school mathematics project, Teacher's Guide for Book
T, Cambridge University Press, London, 1965.
school mathematics project, Book T4, Cambridge
University Press, London, 1965.
school mathematics project, Teacher's Guide for Book
T4, Cambridge University Press, London, 1966.
bibliography
143
TERPSTRA, p., Some notes on the Mathematical Background
of Repetitive Patterns, Oldbourne Press, London, 1961.
thomsen, g., "The Treatment of Elementary Geometry by a
Group-calculus", Mathematical Gazette, 17, No. 224 (Octo-
ber 1933), pp. 230-42.
wells, A. F., The Third Dimension in Chemistry, Clarendon
Press, Oxford, 1956.
YAGLOM, I. m., Geometric Transformations, Random House,
New York, 1962.
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